> 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/tree.md).

# 树

### 树的基本概念

树是一种**非线性**的层次结构。由 n（n≥0）个结点组成的有限集合：

* **根结点**：没有父结点的唯一结点
* **子树**：某结点及其所有后代构成的集合
* **叶子结点**：没有子结点的结点
* **结点的度**：该结点子树的个数
* **树的度**：所有结点度的最大值
* **层次 / 深度**：根为第 1 层，树中最大层次为树的深度

### 二叉树

每个结点最多两个孩子，分**左子树**和**右子树**，顺序不可交换。

常见形态：满二叉树、完全二叉树、斜树。

### 重要性质

* 第 i 层最多 `2^(i-1)` 个结点
* 深度为 k 的二叉树最多 `2^k - 1` 个结点
* 叶子数 `n0 = n2 + 1`（n2 为度为 2 的结点数）
* n 个结点的完全二叉树深度为 `⌊log₂n⌋ + 1`

### 存储方式

* **顺序存储**：用数组，结点 i 的孩子为 2i、2i+1（适合完全二叉树）
* **链式存储**：结点的左右孩子用指针

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

### 树的遍历

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

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

> 相关：二叉树、完全二叉树、二叉排序树、哈夫曼树，见「数据结构」分组其它页面。


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