> 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/huffman-tree.md).

# 哈夫曼树

### 带权路径长度

* 结点的**路径长度**：从根到该结点经过的边数
* 树的**带权路径长度 WPL**：所有叶子结点的（权值 × 路径长度）之和
* **哈夫曼树**：WPL 最小的二叉树，又称最优二叉树

### 构造方法（贪心）

1. 把所有结点放入集合；
2. 每次取出**权值最小的两个**结点，合并成新结点（权值 = 两者之和），放回集合；
3. 重复直到只剩一个结点，即为哈夫曼树的根。

### 示例

权值 1、2、3、4：

```
取 1,2 合并为 3  → {3,3,4}
取 3,3 合并为 6  → {4,6}
取 4,6 合并为 10 → 根为 10
WPL = 1×3 + 2×3 + 3×2 + 4×1 = 19
```

### 优先队列实现

```cpp
priority_queue<int, vector<int>, greater<int>> pq; // 小根堆
for (int i = 0; i < n; i++) pq.push(w[i]);
int ans = 0;
while (pq.size() > 1) {
    int a = pq.top(); pq.pop();
    int b = pq.top(); pq.pop();
    ans += a + b;
    pq.push(a + b);
}
cout << ans << endl;  // 最小 WPL
```


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