> 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/complete-binary-tree.md).

# 完全二叉树

### 定义

若二叉树的深度为 k，前 k-1 层都是满的，且第 k 层的结点都**从左向右连续**排列，则为完全二叉树。

### 重要结论

* 有 n 个结点的完全二叉树，深度为 `⌊log₂n⌋ + 1`。
* 可用**一维数组**顺序存储，下标 1\~n：
  * 左孩子：`2i`
  * 右孩子：`2i + 1`
  * 父结点：`i / 2`
* 叶子结点个数 = `⌈n/2⌉`（即下标大于 `n/2` 的都是叶子）。

### 判断是否为堆的示例

```cpp
// 数组下标从 1 开始，判断小根堆
bool ok = true;
for (int i = 1; i <= n / 2; i++) {
    if (a[2 * i] < a[i]) ok = false;
    if (2 * i + 1 <= n && a[2 * i + 1] < a[i]) ok = false;
}
```

> 满二叉树一定是完全二叉树，完全二叉树不一定是满二叉树。


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