> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/data-structures/binary-tree.md).

# 二叉树

### 基本概念

二叉树是每个结点最多有两个子结点的树，分**左子树**和**右子树**，顺序不能互换。

* **根结点**：最上层结点
* **叶子结点**：没有子结点
* **深度 / 高度**：结点所在层数 / 树的最大层数

### 常见性质

* 第 i 层最多有 `2^(i-1)` 个结点
* 深度为 k 的二叉树最多有 `2^k - 1` 个结点
* 叶子数 n0 与度数为 2 的结点数 n2 满足：`n0 = n2 + 1`

### 存储方式

**顺序存储**（适合完全二叉树）：下标从 1 开始，结点 i 的左孩子为 `2i`，右孩子为 `2i+1`，父结点为 `i/2`。

**链式存储**：

```cpp
struct Node {
    int val;
    Node *left, *right;
    Node(int v) : val(v), left(nullptr), right(nullptr) {}
};
```

### 遍历方式

* **先序**：根 → 左 → 右
* **中序**：左 → 根 → 右
* **后序**：左 → 右 → 根
* **层序**：逐层从左到右（借助队列）

```cpp
void preorder(Node* r) { if (!r) return; cout << r->val; preorder(r->left); preorder(r->right); }
void inorder(Node* r)  { if (!r) return; inorder(r->left); cout << r->val; inorder(r->right); }
void postorder(Node* r){ if (!r) return; postorder(r->left); postorder(r->right); cout << r->val; }
```


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