479 lines
12 KiB
Markdown
479 lines
12 KiB
Markdown
---
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comments: true
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---
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# 二叉树
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「二叉树 Binary Tree」是一种非线性数据结构，代表着祖先与后代之间的派生关系，体现着 “一分为二” 的分治逻辑。类似于链表，二叉树也是以结点为单位存储的，结点包含「值」和两个「指针」。
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=== "Java"
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```java title=""
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/* 链表结点类 */
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class TreeNode {
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int val; // 结点值
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TreeNode left; // 左子结点指针
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TreeNode right; // 右子结点指针
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TreeNode(int x) { val = x; }
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}
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```
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=== "C++"
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```cpp title=""
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/* 链表结点结构体 */
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struct TreeNode {
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int val; // 结点值
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TreeNode *left; // 左子结点指针
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TreeNode *right; // 右子结点指针
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TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
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};
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```
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=== "Python"
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```python title=""
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""" 链表结点类 """
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class TreeNode:
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def __init__(self, val=0, left=None, right=None):
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self.val = val # 结点值
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self.left = left # 左子结点指针
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self.right = right # 右子结点指针
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```
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=== "Go"
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```go title=""
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```
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=== "JavaScript"
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```js title=""
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```
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=== "TypeScript"
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```typescript title=""
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```
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=== "C"
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```c title=""
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```
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=== "C#"
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```csharp title=""
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```
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结点的两个指针分别指向「左子结点 Left Child Node」和「右子结点 Right Child Node」，并且称该结点为两个子结点的「父结点 Parent Node」。给定二叉树某结点，将左子结点以下的树称为该结点的「左子树 Left Subtree」，右子树同理。
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<p align="center"> Fig. 子结点与子树 </p>
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需要注意，父结点、子结点、子树是可以向下递推的。例如，如果将上图的「结点 2」看作父结点，那么其左子结点和右子结点分别为「结点 4」和「结点 5」，左子树和右子树分别为「结点 4 以下的树」和「结点 5 以下的树」。
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## 二叉树常见术语
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「根结点 Root Node」：二叉树最顶层的结点，其没有父结点；
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「叶结点 Leaf Node」：没有子结点的结点，其两个指针都指向 $\text{null}$ ；
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结点「度 Degree」：结点的子结点数量，二叉树中度的范围是 0, 1, 2 ；
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结点「深度 Depth」 ：根结点到该结点的层数；
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结点「高度 Height」：最远叶结点到该结点的层数；
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二叉树「高度」：二叉树中根结点到最远叶结点的层数；
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<p align="center"> Fig. 二叉树的常见术语 </p>
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## 二叉树最佳和最差结构
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当二叉树的每层的结点都被填满时，达到「完美二叉树」；而当所有结点都偏向一边时，二叉树退化为「链表」。
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<p align="center"> Fig. 二叉树的最佳和最差结构 </p>
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在最佳和最差结构下，二叉树的结点数量和高度等性质达到最大（最小）值。
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<div class="center-table" markdown>
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| | 完美二叉树 | 链表 |
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| ----------------------------- | ---------- | ---------- |
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| 二叉树第 $i$ 层的结点数量 | $2^{i-1}$ | $1$ |
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| 高度为 $h$ 的二叉树的结点总数 | $2^h - 1$ | $h$ |
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| 结点总数为 $n$ 的二叉树的高度 | $\log_2 n + 1$ | $n$ |
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</div>
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## 二叉树基本操作
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**初始化二叉树。** 与链表类似，先初始化结点，再构建引用指向（即指针）。
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=== "Java"
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```java title="binary_tree.java"
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// 初始化结点
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TreeNode n1 = new TreeNode(1);
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TreeNode n2 = new TreeNode(2);
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TreeNode n3 = new TreeNode(3);
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TreeNode n4 = new TreeNode(4);
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TreeNode n5 = new TreeNode(5);
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// 构建引用指向（即指针）
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n1.left = n2;
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n1.right = n3;
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n2.left = n4;
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n2.right = n5;
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```
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=== "C++"
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```cpp title="binary_tree.cpp"
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/* 初始化二叉树 */
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// 初始化结点
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TreeNode* n1 = new TreeNode(1);
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TreeNode* n2 = new TreeNode(2);
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TreeNode* n3 = new TreeNode(3);
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TreeNode* n4 = new TreeNode(4);
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TreeNode* n5 = new TreeNode(5);
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// 构建引用指向（即指针）
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n1->left = n2;
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n1->right = n3;
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n2->left = n4;
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n2->right = n5;
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```
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=== "Python"
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```python title="binary_tree.py"
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```
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=== "Go"
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```go title="binary_tree.go"
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```
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=== "JavaScript"
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```js title="binary_tree.js"
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```
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=== "TypeScript"
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```typescript title="binary_tree.ts"
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```
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=== "C"
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```c title="binary_tree.c"
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```
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=== "C#"
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```csharp title="binary_tree.cs"
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```
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**插入与删除结点。** 与链表类似，插入与删除结点都可以通过修改指针实现。
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<p align="center"> Fig. 在二叉树中插入与删除结点 </p>
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=== "Java"
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```java title="binary_tree.java"
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TreeNode P = new TreeNode(0);
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// 在 n1 -> n2 中间插入结点 P
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n1.left = P;
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P.left = n2;
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// 删除结点 P
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n1.left = n2;
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```
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=== "C++"
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```cpp title="binary_tree.cpp"
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/* 插入与删除结点 */
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TreeNode* P = new TreeNode(0);
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// 在 n1 -> n2 中间插入结点 P
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n1->left = P;
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P->left = n2;
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// 删除结点 P
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n1->left = n2;
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```
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=== "Python"
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```python title="binary_tree.py"
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```
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=== "Go"
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```go title="binary_tree.go"
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```
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=== "JavaScript"
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```js title="binary_tree.js"
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```
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=== "TypeScript"
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```typescript title="binary_tree.ts"
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```
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=== "C"
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```c title="binary_tree.c"
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```
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=== "C#"
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```csharp title="binary_tree.cs"
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```
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!!! note
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插入结点会改变二叉树的原有逻辑结构，删除结点往往意味着删除了该结点的所有子树。因此，二叉树中的插入与删除一般都是由一套操作配合完成的，这样才能实现有意义的操作。
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## 二叉树遍历
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非线性数据结构的遍历操作比线性数据结构更加复杂，往往需要使用搜索算法来实现。常见的二叉树遍历方式有层序遍历、前序遍历、中序遍历、后序遍历。
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### 层序遍历
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「层序遍历 Hierarchical-Order Traversal」从顶至底、一层一层地遍历二叉树，并在每层中按照从左到右的顺序访问结点。
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层序遍历本质上是「广度优先搜索 Breadth-First Traversal」，其体现着一种 “一圈一圈向外” 的层进遍历方式。
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<p align="center"> Fig. 二叉树的层序遍历 </p>
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广度优先遍历一般借助「队列」来实现。队列的规则是 “先进先出” ，广度优先遍历的规则是 ”一层层平推“ ，两者背后的思想是一致的。
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=== "Java"
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```java title="binary_tree_bfs.java"
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/* 层序遍历 */
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List<Integer> hierOrder(TreeNode root) {
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// 初始化队列，加入根结点
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Queue<TreeNode> queue = new LinkedList<>() {{ add(root); }};
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// 初始化一个列表，用于保存遍历序列
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List<Integer> list = new ArrayList<>();
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while (!queue.isEmpty()) {
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TreeNode node = queue.poll(); // 队列出队
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list.add(node.val); // 保存结点值
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if (node.left != null)
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queue.offer(node.left); // 左子结点入队
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if (node.right != null)
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queue.offer(node.right); // 右子结点入队
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}
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return list;
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}
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```
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=== "C++"
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```cpp title="binary_tree_bfs.cpp"
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/* 层序遍历 */
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vector<int> hierOrder(TreeNode* root) {
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// 初始化队列，加入根结点
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queue<TreeNode*> queue;
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queue.push(root);
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// 初始化一个列表，用于保存遍历序列
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vector<int> vec;
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while (!queue.empty()) {
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TreeNode* node = queue.front();
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queue.pop(); // 队列出队
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vec.push_back(node->val); // 保存结点
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if (node->left != nullptr)
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queue.push(node->left); // 左子结点入队
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if (node->right != nullptr)
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queue.push(node->right); // 右子结点入队
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}
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return vec;
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}
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```
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=== "Python"
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```python title="binary_tree_bfs.py"
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```
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=== "Go"
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```go title="binary_tree_bfs.go"
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```
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=== "JavaScript"
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```js title="binary_tree_bfs.js"
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```
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=== "TypeScript"
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```typescript title="binary_tree_bfs.ts"
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```
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=== "C"
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```c title="binary_tree_bfs.c"
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```
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=== "C#"
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```csharp title="binary_tree_bfs.cs"
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```
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### 前序、中序、后序遍历
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相对地，前、中、后序遍历皆属于「深度优先遍历 Depth-First Traversal」，其体现着一种 “先走到尽头，再回头继续” 的回溯遍历方式。
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如下图所示，左侧是深度优先遍历的的示意图，右上方是对应的递归实现代码。深度优先遍历就像是绕着整个二叉树的外围 “走” 一圈，走的过程中，在每个结点都会遇到三个位置，分别对应前序遍历、中序遍历、后序遍历。
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<p align="center"> Fig. 二叉树的前 / 中 / 后序遍历 </p>
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<div class="center-table" markdown>
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| 位置 | 含义 | 此处访问结点时对应 |
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| ---------- | ------------------------------------ | ----------------------------- |
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| 橙色圆圈处 | 刚进入此结点，即将访问该结点的左子树 | 前序遍历 Pre-Order Traversal |
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| 蓝色圆圈处 | 已访问完左子树，即将访问右子树 | 中序遍历 In-Order Traversal |
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| 紫色圆圈处 | 已访问完左子树和右子树，即将返回 | 后序遍历 Post-Order Traversal |
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</div>
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=== "Java"
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```java title="binary_tree_dfs.java"
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/* 前序遍历 */
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void preOrder(TreeNode root) {
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if (root == null) return;
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// 访问优先级：根结点 -> 左子树 -> 右子树
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list.add(root.val);
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preOrder(root.left);
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preOrder(root.right);
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}
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/* 中序遍历 */
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void inOrder(TreeNode root) {
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if (root == null) return;
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// 访问优先级：左子树 -> 根结点 -> 右子树
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inOrder(root.left);
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list.add(root.val);
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inOrder(root.right);
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}
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/* 后序遍历 */
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void postOrder(TreeNode root) {
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if (root == null) return;
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// 访问优先级：左子树 -> 右子树 -> 根结点
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postOrder(root.left);
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postOrder(root.right);
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list.add(root.val);
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}
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```
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=== "C++"
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```cpp title="binary_tree_dfs.cpp"
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/* 前序遍历 */
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void preOrder(TreeNode* root) {
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if (root == nullptr) return;
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// 访问优先级：根结点 -> 左子树 -> 右子树
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vec.push_back(root->val);
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preOrder(root->left);
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preOrder(root->right);
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}
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/* 中序遍历 */
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void inOrder(TreeNode* root) {
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if (root == nullptr) return;
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// 访问优先级：左子树 -> 根结点 -> 右子树
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inOrder(root->left);
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vec.push_back(root->val);
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inOrder(root->right);
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}
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/* 后序遍历 */
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void postOrder(TreeNode* root) {
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if (root == nullptr) return;
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// 访问优先级：左子树 -> 右子树 -> 根结点
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postOrder(root->left);
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postOrder(root->right);
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vec.push_back(root->val);
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}
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```
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=== "Python"
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```python title="binary_tree_dfs.py"
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```
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=== "Go"
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```go title="binary_tree_dfs.go"
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```
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=== "JavaScript"
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```js title="binary_tree_dfs.js"
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```
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=== "TypeScript"
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```typescript title="binary_tree_dfs.ts"
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```
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=== "C"
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```c title="binary_tree_dfs.c"
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```
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=== "C#"
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```csharp title="binary_tree_dfs.cs"
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```
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!!! note
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使用循环一样可以实现前、中、后序遍历，但代码相对繁琐，有兴趣的同学可以自行实现。
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