2282 lines
118 KiB
HTML
2282 lines
118 KiB
HTML
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<!doctype html>
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<html lang="zh" class="no-js">
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<head>
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<meta name="viewport" content="width=device-width,initial-scale=1">
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<meta name="description" content="Your first book to learn Data Structure And Algorithm.">
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<meta name="author" content="Krahets">
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<link rel="canonical" href="https://www.hello-algo.com/chapter_tree/avl_tree/">
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<title>AVL 树 * - Hello 算法</title>
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<script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce((e,_)=>(e<<5)-e+_.charCodeAt(0),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
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</head>
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<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="">
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<script>var palette=__md_get("__palette");if(palette&&"object"==typeof palette.color)for(var key of Object.keys(palette.color))document.body.setAttribute("data-md-color-"+key,palette.color[key])</script>
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跳转至
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<header class="md-header" data-md-component="header">
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<nav class="md-header__inner md-grid" aria-label="页眉">
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<a href="../.." title="Hello 算法" class="md-header__button md-logo" aria-label="Hello 算法" data-md-component="logo">
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<img src="../../assets/images/logo.png" alt="logo">
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</a>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3V6m0 5h18v2H3v-2m0 5h18v2H3v-2Z"/></svg>
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<div class="md-header__title" data-md-component="header-title">
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<div class="md-header__topic">
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<span class="md-ellipsis">
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Hello 算法
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</span>
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</div>
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<div class="md-header__topic" data-md-component="header-topic">
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<span class="md-ellipsis">
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AVL 树 *
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</span>
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</div>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m17.75 4.09-2.53 1.94.91 3.06-2.63-1.81-2.63 1.81.91-3.06-2.53-1.94L12.44 4l1.06-3 1.06 3 3.19.09m3.5 6.91-1.64 1.25.59 1.98-1.7-1.17-1.7 1.17.59-1.98L15.75 11l2.06-.05L18.5 9l.69 1.95 2.06.05m-2.28 4.95c.83-.08 1.72 1.1 1.19 1.85-.32.45-.66.87-1.08 1.27C15.17 23 8.84 23 4.94 19.07c-3.91-3.9-3.91-10.24 0-14.14.4-.4.82-.76 1.27-1.08.75-.53 1.93.36 1.85 1.19-.27 2.86.69 5.83 2.89 8.02a9.96 9.96 0 0 0 8.02 2.89m-1.64 2.02a12.08 12.08 0 0 1-7.8-3.47c-2.17-2.19-3.33-5-3.49-7.82-2.81 3.14-2.7 7.96.31 10.98 3.02 3.01 7.84 3.12 10.98.31Z"/></svg>
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<input type="text" class="md-search__input" name="query" aria-label="搜索" placeholder="搜索" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
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<div class="md-search-result__meta">
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正在初始化搜索引擎
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</div>
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<ol class="md-search-result__list"></ol>
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<div class="md-header__source">
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<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
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krahets/hello-algo
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<a href="../.." title="Hello 算法" class="md-nav__button md-logo" aria-label="Hello 算法" data-md-component="logo">
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</a>
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Hello 算法
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<a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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|
|||
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|
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|
|
<label class="md-nav__link" for="__nav_5">
|
|||
|
|
数组与链表
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="数组与链表" data-md-level="1">
|
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|
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<label class="md-nav__title" for="__nav_5">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
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数组与链表
|
|||
|
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</label>
|
|||
|
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<ul class="md-nav__list" data-md-scrollfix>
|
|||
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|
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
|
|||
|
|
数组（Array）
|
|||
|
|
</a>
|
|||
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</li>
|
|||
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|||
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
|
|||
|
|
链表（LinkedList）
|
|||
|
|
</a>
|
|||
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</li>
|
|||
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<li class="md-nav__item">
|
|||
|
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<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
|
|||
|
|
列表（List）
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
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</li>
|
|||
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|
|||
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|
|||
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|
|||
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|
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</ul>
|
|||
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</nav>
|
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|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
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|
|||
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|
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|
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<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_6" type="checkbox" id="__nav_6" >
|
|||
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|
|||
|
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|
|||
|
|
<label class="md-nav__link" for="__nav_6">
|
|||
|
|
栈与队列
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="栈与队列" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_6">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
栈与队列
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
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|
|||
|
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|
|||
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
|
|||
|
|
栈（Stack）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
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|
|||
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|
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
|
|||
|
|
队列（Queue）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
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|
|
|||
|
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|
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|
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|
|||
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|
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|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
|
|||
|
|
双向队列（Deque）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
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|
|||
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|
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|
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|
|||
|
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|
|||
|
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<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
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|
|||
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|
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|
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|
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|
|||
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|
|||
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|
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|
|||
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|
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|
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_7" type="checkbox" id="__nav_7" >
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
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|
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_7">
|
|||
|
|
散列表
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="散列表" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_7">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
散列表
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
|
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|
|||
|
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|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
|
|||
|
|
哈希表（HashMap）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
|||
|
|
哈希冲突处理
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
|
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|
|||
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|
|||
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|
|||
|
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_hashing/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|
|||
|
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|
|||
|
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_8" type="checkbox" id="__nav_8" checked>
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_8">
|
|||
|
|
二叉树
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="二叉树" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_8">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
二叉树
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../binary_tree/" class="md-nav__link">
|
|||
|
|
二叉树（Binary Tree）
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../binary_tree_types/" class="md-nav__link">
|
|||
|
|
二叉树常见类型
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../binary_search_tree/" class="md-nav__link">
|
|||
|
|
二叉搜索树
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
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|
|||
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|
|||
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|
|||
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|
|||
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|
|||
|
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|
|||
|
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|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--active">
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="toc" type="checkbox" id="__toc">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link md-nav__link--active" for="__toc">
|
|||
|
|
AVL 树 *
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<a href="./" class="md-nav__link md-nav__link--active">
|
|||
|
|
AVL 树 *
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__title" for="__toc">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
目录
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_1" class="md-nav__link">
|
|||
|
|
AVL 树常见术语
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="AVL 树常见术语">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_1" class="md-nav__link">
|
|||
|
|
结点高度
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_2" class="md-nav__link">
|
|||
|
|
结点平衡因子
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_2" class="md-nav__link">
|
|||
|
|
AVL 树旋转
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="AVL 树旋转">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-1-" class="md-nav__link">
|
|||
|
|
Case 1 - 右旋
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-2-" class="md-nav__link">
|
|||
|
|
Case 2 - 左旋
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-3-" class="md-nav__link">
|
|||
|
|
Case 3 - 先左后右
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-4-" class="md-nav__link">
|
|||
|
|
Case 4 - 先右后左
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_3" class="md-nav__link">
|
|||
|
|
旋转的选择
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_3" class="md-nav__link">
|
|||
|
|
AVL 树常用操作
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="AVL 树常用操作">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_4" class="md-nav__link">
|
|||
|
|
插入结点
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_5" class="md-nav__link">
|
|||
|
|
删除结点
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_6" class="md-nav__link">
|
|||
|
|
查找结点
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_4" class="md-nav__link">
|
|||
|
|
AVL 树典型应用
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_9" type="checkbox" id="__nav_9" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_9">
|
|||
|
|
查找算法
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="查找算法" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_9">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
查找算法
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
|
|||
|
|
线性查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
|
|||
|
|
二分查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
|
|||
|
|
哈希查找
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_searching/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_10" type="checkbox" id="__nav_10" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__link" for="__nav_10">
|
|||
|
|
排序算法
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
</label>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="排序算法" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_10">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
排序算法
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
|
|||
|
|
排序简介
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
|
|||
|
|
冒泡排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
|
|||
|
|
插入排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
|
|||
|
|
快速排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
|
|||
|
|
归并排序
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="../../chapter_sorting/summary/" class="md-nav__link">
|
|||
|
|
小结
|
|||
|
|
</a>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<li class="md-nav__item md-nav__item--section md-nav__item--nested">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<input class="md-nav__toggle md-toggle" data-md-toggle="__nav_11" type="checkbox" id="__nav_11" >
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-nav__link md-nav__link--index ">
|
|||
|
|
<a href="../../chapter_reference/">参考文献</a>
|
|||
|
|
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="参考文献" data-md-level="1">
|
|||
|
|
<label class="md-nav__title" for="__nav_11">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
参考文献
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
|
|||
|
|
<div class="md-sidebar__scrollwrap">
|
|||
|
|
<div class="md-sidebar__inner">
|
|||
|
|
|
|||
|
|
|
|||
|
|
<nav class="md-nav md-nav--secondary" aria-label="目录">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<label class="md-nav__title" for="__toc">
|
|||
|
|
<span class="md-nav__icon md-icon"></span>
|
|||
|
|
目录
|
|||
|
|
</label>
|
|||
|
|
<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_1" class="md-nav__link">
|
|||
|
|
AVL 树常见术语
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="AVL 树常见术语">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_1" class="md-nav__link">
|
|||
|
|
结点高度
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_2" class="md-nav__link">
|
|||
|
|
结点平衡因子
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
</nav>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_2" class="md-nav__link">
|
|||
|
|
AVL 树旋转
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="AVL 树旋转">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-1-" class="md-nav__link">
|
|||
|
|
Case 1 - 右旋
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-2-" class="md-nav__link">
|
|||
|
|
Case 2 - 左旋
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-3-" class="md-nav__link">
|
|||
|
|
Case 3 - 先左后右
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#case-4-" class="md-nav__link">
|
|||
|
|
Case 4 - 先右后左
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
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|||
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|
<li class="md-nav__item">
|
|||
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<a href="#_3" class="md-nav__link">
|
|||
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旋转的选择
|
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|
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</a>
|
|||
|
|
|
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|
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</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
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</nav>
|
|||
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|||
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</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_3" class="md-nav__link">
|
|||
|
|
AVL 树常用操作
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
<nav class="md-nav" aria-label="AVL 树常用操作">
|
|||
|
|
<ul class="md-nav__list">
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_4" class="md-nav__link">
|
|||
|
|
插入结点
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_5" class="md-nav__link">
|
|||
|
|
删除结点
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#_6" class="md-nav__link">
|
|||
|
|
查找结点
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
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|
|
|||
|
|
</ul>
|
|||
|
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</nav>
|
|||
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|||
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|
</li>
|
|||
|
|
|
|||
|
|
<li class="md-nav__item">
|
|||
|
|
<a href="#avl_4" class="md-nav__link">
|
|||
|
|
AVL 树典型应用
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
</li>
|
|||
|
|
|
|||
|
|
</ul>
|
|||
|
|
|
|||
|
|
</nav>
|
|||
|
|
</div>
|
|||
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|
</div>
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<div class="md-content" data-md-component="content">
|
|||
|
|
<article class="md-content__inner md-typeset">
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<a href="https://github.com/krahets/hello-algo/tree/master/docs/chapter_tree/avl_tree.md" title="编辑此页" class="md-content__button md-icon">
|
|||
|
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|
|||
|
|
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20.71 7.04c.39-.39.39-1.04 0-1.41l-2.34-2.34c-.37-.39-1.02-.39-1.41 0l-1.84 1.83 3.75 3.75M3 17.25V21h3.75L17.81 9.93l-3.75-3.75L3 17.25Z"/></svg>
|
|||
|
|
</a>
|
|||
|
|
|
|||
|
|
|
|||
|
|
<h1 id="avl">AVL 树 *<a class="headerlink" href="#avl" title="Permanent link">¶</a></h1>
|
|||
|
|
<p>在「二叉搜索树」章节中提到，在进行多次插入与删除操作后，二叉搜索树可能会退化为链表。此时所有操作的时间复杂度都会由 <span class="arithmatex">\(O(\log n)\)</span> 劣化至 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
|||
|
|
<p>如下图所示，执行两步删除结点后，该二叉搜索树就会退化为链表。</p>
|
|||
|
|
<p><img alt="degradation_from_removing_node" src="../avl_tree.assets/degradation_from_removing_node.png" /></p>
|
|||
|
|
<p>再比如，在以下完美二叉树中插入两个结点后，树严重向左偏斜，查找操作的时间复杂度也随之发生劣化。</p>
|
|||
|
|
<p><img alt="degradation_from_inserting_node" src="../avl_tree.assets/degradation_from_inserting_node.png" /></p>
|
|||
|
|
<p>G. M. Adelson-Velsky 和 E. M. Landis 在其 1962 年发表的论文 "An algorithm for the organization of information" 中提出了「AVL 树」。<strong>论文中描述了一系列操作，使得在不断添加与删除结点后，AVL 树仍然不会发生退化</strong>，进而使得各种操作的时间复杂度均能保持在 <span class="arithmatex">\(O(\log n)\)</span> 级别。</p>
|
|||
|
|
<p>换言之，在频繁增删查改的使用场景中，AVL 树可始终保持很高的数据增删查改效率，具有很好的应用价值。</p>
|
|||
|
|
<h2 id="avl_1">AVL 树常见术语<a class="headerlink" href="#avl_1" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>「AVL 树」既是「二叉搜索树」又是「平衡二叉树」，同时满足这两种二叉树的所有性质，因此又被称为「平衡二叉搜索树」。</p>
|
|||
|
|
<h3 id="_1">结点高度<a class="headerlink" href="#_1" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>在 AVL 树的操作中，需要获取结点「高度 Height」，所以给 AVL 树的结点类添加 <code>height</code> 变量。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="1:8"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* AVL 树结点类 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// 结点值</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span><span class="w"> </span><span class="c1">// 结点高度</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// 左子结点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// 右子结点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="kd">public</span><span class="w"> </span><span class="nf">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>「结点高度」是最远叶结点到该结点的距离，即走过的「边」的数量。需要特别注意，<strong>叶结点的高度为 0 ，空结点的高度为 -1</strong> 。我们封装两个工具函数，分别用于获取与更新结点的高度。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="2:8"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 获取结点高度 */</span><span class="w"> </span>
|
|||
|
|
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w"> </span><span class="c1">// 空结点高度为 -1 ，叶结点高度为 0</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">height</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a>
|
|||
|
|
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="cm">/* 更新结点高度 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w"> </span><span class="c1">// 结点高度等于最高子树高度 + 1</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span>
|
|||
|
|
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
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</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="_2">结点平衡因子<a class="headerlink" href="#_2" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>结点的「平衡因子 Balance Factor」是 <strong>结点的左子树高度减去右子树高度</strong>，并定义空结点的平衡因子为 0 。同样地，我们将获取结点平衡因子封装成函数，以便后续使用。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="3:8"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label></div>
|
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<div class="tabbed-content">
|
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<div class="tabbed-block">
|
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<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 获取结点平衡因子 */</span><span class="w"> </span>
|
|||
|
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<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">balanceFactor</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w"> </span><span class="c1">// 空结点平衡因子为 0</span><span class="w"></span>
|
|||
|
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<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w"> </span><span class="c1">// 结点平衡因子 = 左子树高度 - 右子树高度</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
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<div class="tabbed-block">
|
|||
|
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<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<div class="admonition note">
|
|||
|
|
<p class="admonition-title">Note</p>
|
|||
|
|
<p>设平衡因子为 <span class="arithmatex">\(f\)</span> ，则一棵 AVL 树的任意结点的平衡因子皆满足 <span class="arithmatex">\(-1 \le f \le 1\)</span> 。</p>
|
|||
|
|
</div>
|
|||
|
|
<h2 id="avl_2">AVL 树旋转<a class="headerlink" href="#avl_2" title="Permanent link">¶</a></h2>
|
|||
|
|
<p>AVL 树的独特之处在于「旋转 Rotation」的操作，其可 <strong>在不影响二叉树中序遍历序列的前提下，使失衡结点重新恢复平衡。</strong> 换言之，旋转操作既可以使树保持为「二叉搜索树」，也可以使树重新恢复为「平衡二叉树」。</p>
|
|||
|
|
<p>我们将平衡因子的绝对值 <span class="arithmatex">\(> 1\)</span> 的结点称为「失衡结点」。根据结点的失衡情况，旋转操作分为 <strong>右旋、左旋、先右旋后左旋、先左旋后右旋</strong>，接下来我们来一起来看看它们是如何操作的。</p>
|
|||
|
|
<h3 id="case-1-">Case 1 - 右旋<a class="headerlink" href="#case-1-" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>如下图所示（结点下方为「平衡因子」），从底至顶看，二叉树中首个失衡结点是 <strong>结点 3</strong> 。我们聚焦在以该失衡结点为根结点的子树上，将该结点记为 <code>node</code> ，将其左子节点记为 <code>child</code> ，执行「右旋」操作。完成右旋后，该子树已经恢复平衡，并且仍然为二叉搜索树。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="4:4"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Step 1</label><label for="__tabbed_4_2">Step 2</label><label for="__tabbed_4_3">Step 3</label><label for="__tabbed_4_4">Step 4</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="right_rotate_step1" src="../avl_tree.assets/right_rotate_step1.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="right_rotate_step2" src="../avl_tree.assets/right_rotate_step2.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="right_rotate_step3" src="../avl_tree.assets/right_rotate_step3.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<p><img alt="right_rotate_step4" src="../avl_tree.assets/right_rotate_step4.png" /></p>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<p>进而，如果结点 <code>child</code> 本身有右子结点（记为 <code>grandChild</code>），则需要在「右旋」中添加一步：将 <code>grandChild</code> 作为 <code>node</code> 的左子结点。</p>
|
|||
|
|
<p><img alt="right_rotate_with_grandchild" src="../avl_tree.assets/right_rotate_with_grandchild.png" /></p>
|
|||
|
|
<p>“向右旋转” 是一种形象化的说法，实际需要通过修改结点指针实现，代码如下所示。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="5:8"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JavaScript</label><label for="__tabbed_5_6">TypeScript</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* 右旋操作 */</span><span class="w"> </span>
|
|||
|
|
<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">rightRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="na">right</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w"> </span><span class="c1">// 以 child 为原点，将 node 向右旋转</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w"> </span><span class="c1">// 更新结点高度</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w"> </span><span class="c1">// 返回旋转后子树的根节点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="case-2-">Case 2 - 左旋<a class="headerlink" href="#case-2-" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>类似地，如果将取上述失衡二叉树的 “镜像” ，那么则需要「左旋」操作。观察发现，<strong>「左旋」和「右旋」操作是镜像对称的，两者对应解决的两种失衡情况也是对称的</strong>。</p>
|
|||
|
|
<p><img alt="left_rotate_with_grandchild" src="../avl_tree.assets/left_rotate_with_grandchild.png" /></p>
|
|||
|
|
<p>根据对称性，我们可以很方便地从「右旋」推导出「左旋」。具体地，把所有的 <code>left</code> 替换为 <code>right</code> 、所有的 <code>right</code> 替换为 <code>left</code> 即可。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="6:8"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JavaScript</label><label for="__tabbed_6_6">TypeScript</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label></div>
|
|||
|
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<div class="tabbed-content">
|
|||
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* 左旋操作 */</span><span class="w"> </span>
|
|||
|
|
<a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">private</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="nf">leftRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="na">left</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w"> </span><span class="c1">// 以 child 为原点，将 node 向左旋转</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w"> </span><span class="c1">// 更新结点高度</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w"> </span><span class="c1">// 返回旋转后子树的根节点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="case-3-">Case 3 - 先左后右<a class="headerlink" href="#case-3-" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>对于下图的失衡结点 3 ，<strong>单一使用左旋或右旋都无法使子树恢复平衡</strong>，此时需要「先左旋后右旋」，即先对 <code>child</code> 执行「左旋」，再对 <code>node</code> 执行「右旋」。</p>
|
|||
|
|
<p><img alt="left_right_rotate" src="../avl_tree.assets/left_right_rotate.png" /></p>
|
|||
|
|
<h3 id="case-4-">Case 4 - 先右后左<a class="headerlink" href="#case-4-" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>同理，取以上失衡二叉树的镜像，则需要「先右旋后左旋」，即先对 <code>child</code> 执行「右旋」，然后对 <code>node</code> 执行「左旋」。</p>
|
|||
|
|
<p><img alt="right_left_rotate" src="../avl_tree.assets/right_left_rotate.png" /></p>
|
|||
|
|
<h3 id="_3">旋转的选择<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>下图描述的四种失衡情况与上述 Cases 一一对应，分别采用右旋、左旋、先右后左、先左后右的旋转组合。</p>
|
|||
|
|
<p><img alt="rotation_cases" src="../avl_tree.assets/rotation_cases.png" /></p>
|
|||
|
|
<p>具体地，需要使用 <strong>失衡结点的平衡因子、较高一侧子结点的平衡因子</strong> 来确定失衡结点属于上图中的哪种情况。</p>
|
|||
|
|
<div class="center-table">
|
|||
|
|
<table>
|
|||
|
|
<thead>
|
|||
|
|
<tr>
|
|||
|
|
<th>失衡结点的平衡因子</th>
|
|||
|
|
<th>子结点的平衡因子</th>
|
|||
|
|
<th>应采用的旋转方法</th>
|
|||
|
|
</tr>
|
|||
|
|
</thead>
|
|||
|
|
<tbody>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(>0\)</span> （即左偏树）</td>
|
|||
|
|
<td><span class="arithmatex">\(\geq 0\)</span></td>
|
|||
|
|
<td>右旋</td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(>0\)</span> （即左偏树）</td>
|
|||
|
|
<td><span class="arithmatex">\(<0\)</span></td>
|
|||
|
|
<td>先左旋后右旋</td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(<0\)</span> （即右偏树）</td>
|
|||
|
|
<td><span class="arithmatex">\(\leq 0\)</span></td>
|
|||
|
|
<td>左旋</td>
|
|||
|
|
</tr>
|
|||
|
|
<tr>
|
|||
|
|
<td><span class="arithmatex">\(<0\)</span> （即右偏树）</td>
|
|||
|
|
<td><span class="arithmatex">\(>0\)</span></td>
|
|||
|
|
<td>先右旋后左旋</td>
|
|||
|
|
</tr>
|
|||
|
|
</tbody>
|
|||
|
|
</table>
|
|||
|
|
</div>
|
|||
|
|
<p>根据以上规则，我们将旋转操作封装成一个函数。至此，<strong>我们可以使用此函数来旋转各种失衡情况，使失衡结点重新恢复平衡</strong>。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="7:8"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JavaScript</label><label for="__tabbed_7_6">TypeScript</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* 执行旋转操作，使该子树重新恢复平衡 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w"> </span><span class="c1">// 获取结点 node 的平衡因子</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w"> </span><span class="c1">// 左偏树</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w"> </span><span class="c1">// 右旋</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w"> </span><span class="c1">// 先左旋后右旋</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w"> </span><span class="c1">// 右偏树</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-19" name="__codelineno-40-19" href="#__codelineno-40-19"></a><span class="w"> </span><span class="c1">// 左旋</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-20" name="__codelineno-40-20" href="#__codelineno-40-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-21" name="__codelineno-40-21" href="#__codelineno-40-21"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-22" name="__codelineno-40-22" href="#__codelineno-40-22"></a><span class="w"> </span><span class="c1">// 先右旋后左旋</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-23" name="__codelineno-40-23" href="#__codelineno-40-23"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-24" name="__codelineno-40-24" href="#__codelineno-40-24"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-25" name="__codelineno-40-25" href="#__codelineno-40-25"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-26" name="__codelineno-40-26" href="#__codelineno-40-26"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-27" name="__codelineno-40-27" href="#__codelineno-40-27"></a><span class="w"> </span><span class="c1">// 平衡树，无需旋转，直接返回</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-28" name="__codelineno-40-28" href="#__codelineno-40-28"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-40-29" name="__codelineno-40-29" href="#__codelineno-40-29"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
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|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
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|
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|
|||
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|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a>
|
|||
|
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</code></pre></div>
|
|||
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|
</div>
|
|||
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a>
|
|||
|
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</code></pre></div>
|
|||
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|
</div>
|
|||
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<div class="tabbed-block">
|
|||
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<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a>
|
|||
|
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</code></pre></div>
|
|||
|
|
</div>
|
|||
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<div class="tabbed-block">
|
|||
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|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
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</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h2 id="avl_3">AVL 树常用操作<a class="headerlink" href="#avl_3" title="Permanent link">¶</a></h2>
|
|||
|
|
<h3 id="_4">插入结点<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>「AVL 树」的结点插入操作与「二叉搜索树」主体类似。不同的是，在插入结点后，从该结点到根结点的路径上会出现一系列「失衡结点」。所以，<strong>我们需要从该结点开始，从底至顶地执行旋转操作，使所有失衡结点恢复平衡</strong>。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="8:8"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JavaScript</label><label for="__tabbed_8_6">TypeScript</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label></div>
|
|||
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<div class="tabbed-content">
|
|||
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<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 插入结点 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a>
|
|||
|
|
<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="cm">/* 递归插入结点（辅助函数） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w"> </span><span class="cm">/* 1. 查找插入位置，并插入结点 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w"> </span><span class="k">else</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// 重复结点不插入，直接返回</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// 更新结点高度</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w"> </span><span class="cm">/* 2. 执行旋转操作，使该子树重新恢复平衡 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-19" name="__codelineno-48-19" href="#__codelineno-48-19"></a><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-20" name="__codelineno-48-20" href="#__codelineno-48-20"></a><span class="w"> </span><span class="c1">// 返回子树的根节点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-21" name="__codelineno-48-21" href="#__codelineno-48-21"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-48-22" name="__codelineno-48-22" href="#__codelineno-48-22"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="_5">删除结点<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>「AVL 树」删除结点操作与「二叉搜索树」删除结点操作总体相同。类似地，<strong>在删除结点后，也需要从底至顶地执行旋转操作，使所有失衡结点恢复平衡</strong>。</p>
|
|||
|
|
<div class="tabbed-set tabbed-alternate" data-tabs="9:8"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JavaScript</label><label for="__tabbed_9_6">TypeScript</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label></div>
|
|||
|
|
<div class="tabbed-content">
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* 删除结点 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a>
|
|||
|
|
<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="cm">/* 递归删除结点（辅助函数） */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w"> </span><span class="cm">/* 1. 查找结点，并删除之 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-14" name="__codelineno-56-14" href="#__codelineno-56-14"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-15" name="__codelineno-56-15" href="#__codelineno-56-15"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-17" name="__codelineno-56-17" href="#__codelineno-56-17"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-18" name="__codelineno-56-18" href="#__codelineno-56-18"></a><span class="w"> </span><span class="c1">// 子结点数量 = 0 ，直接删除 node 并返回</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-19" name="__codelineno-56-19" href="#__codelineno-56-19"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-20" name="__codelineno-56-20" href="#__codelineno-56-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-21" name="__codelineno-56-21" href="#__codelineno-56-21"></a><span class="w"> </span><span class="c1">// 子结点数量 = 1 ，直接删除 node</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-22" name="__codelineno-56-22" href="#__codelineno-56-22"></a><span class="w"> </span><span class="k">else</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-23" name="__codelineno-56-23" href="#__codelineno-56-23"></a><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-24" name="__codelineno-56-24" href="#__codelineno-56-24"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-25" name="__codelineno-56-25" href="#__codelineno-56-25"></a><span class="w"> </span><span class="c1">// 子结点数量 = 2 ，则将中序遍历的下个结点删除，并用该结点替换当前结点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-26" name="__codelineno-56-26" href="#__codelineno-56-26"></a><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minNode</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-27" name="__codelineno-56-27" href="#__codelineno-56-27"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">,</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="na">val</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-28" name="__codelineno-56-28" href="#__codelineno-56-28"></a><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="na">val</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-29" name="__codelineno-56-29" href="#__codelineno-56-29"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-30" name="__codelineno-56-30" href="#__codelineno-56-30"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-31" name="__codelineno-56-31" href="#__codelineno-56-31"></a><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// 更新结点高度</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-32" name="__codelineno-56-32" href="#__codelineno-56-32"></a><span class="w"> </span><span class="cm">/* 2. 执行旋转操作，使该子树重新恢复平衡 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-33" name="__codelineno-56-33" href="#__codelineno-56-33"></a><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-34" name="__codelineno-56-34" href="#__codelineno-56-34"></a><span class="w"> </span><span class="c1">// 返回子树的根节点</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-35" name="__codelineno-56-35" href="#__codelineno-56-35"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-36" name="__codelineno-56-36" href="#__codelineno-56-36"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-37" name="__codelineno-56-37" href="#__codelineno-56-37"></a>
|
|||
|
|
<a id="__codelineno-56-38" name="__codelineno-56-38" href="#__codelineno-56-38"></a><span class="cm">/* 获取最小结点 */</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-39" name="__codelineno-56-39" href="#__codelineno-56-39"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">minNode</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-40" name="__codelineno-56-40" href="#__codelineno-56-40"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-41" name="__codelineno-56-41" href="#__codelineno-56-41"></a><span class="w"> </span><span class="c1">// 循环访问左子结点，直到叶结点时为最小结点，跳出</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-42" name="__codelineno-56-42" href="#__codelineno-56-42"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-43" name="__codelineno-56-43" href="#__codelineno-56-43"></a><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-44" name="__codelineno-56-44" href="#__codelineno-56-44"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-45" name="__codelineno-56-45" href="#__codelineno-56-45"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"></span>
|
|||
|
|
<a id="__codelineno-56-46" name="__codelineno-56-46" href="#__codelineno-56-46"></a><span class="p">}</span><span class="w"></span>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
<div class="tabbed-block">
|
|||
|
|
<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a>
|
|||
|
|
</code></pre></div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
</div>
|
|||
|
|
<h3 id="_6">查找结点<a class="headerlink" href="#_6" title="Permanent link">¶</a></h3>
|
|||
|
|
<p>「AVL 树」的结点查找操作与「二叉搜索树」一致，在此不再赘述。</p>
|
|||
|
|
<h2 id="avl_4">AVL 树典型应用<a class="headerlink" href="#avl_4" title="Permanent link">¶</a></h2>
|
|||
|
|
<ul>
|
|||
|
|
<li>组织存储大型数据，适用于高频查找、低频增删场景</li>
|
|||
|
|
<li>用于建立数据库中的索引系统；</li>
|
|||
|
|
</ul>
|
|||
|
|
<div class="admonition question">
|
|||
|
|
<p class="admonition-title">为什么红黑树比 AVL 树更受欢迎？</p>
|
|||
|
|
<p>红黑树的平衡条件相对宽松，因此在红黑树中插入与删除结点所需的旋转操作相对更少，结点增删操作相比 AVL 树的效率更高。</p>
|
|||
|
|
</div>
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
|
|||
|
|
<h2 id="__comments">评论</h2>
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|||
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var theme = palette.color.scheme === "slate" ? "dark" : "light"
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|
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var frame = document.querySelector(".giscus-frame")
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{ giscus: { setConfig: { theme } } },
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"https://giscus.app"
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