> 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-search-tree.md).

# 二叉排序树

### 定义（BST）

二叉排序树（二叉查找树）满足：**左子树所有结点 < 根 < 右子树所有结点**，且左右子树也都是 BST。

### 作用

支持快速查找、插入、删除，平均时间复杂度 O(log n)。

### 查找

```cpp
struct Node { int val; Node *left, *right; };

Node* find(Node* root, int x) {
    if (!root) return nullptr;
    if (root->val == x) return root;
    if (x < root->val) return find(root->left, x);
    else                return find(root->right, x);
}
```

### 插入

```cpp
Node* insert(Node* root, int x) {
    if (!root) return new Node{x, nullptr, nullptr};
    if (x < root->val) root->left = insert(root->left, x);
    else if (x > root->val) root->right = insert(root->right, x);
    return root;
}
```

### 中序遍历有序

对 BST 进行**中序遍历**，得到的序列是从小到大有序的，这是重要性质。

> 极端情况下 BST 会退化成链表，复杂度变为 O(n)，于是引入平衡二叉树（如 AVL）。


---

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