哈夫曼树是一种带权路径最小的树，运用于压缩编码算法。

带权路径又称WPL，是所有非叶子节点的路径之和，路径=深度*权值

哈夫曼树的节点个数=2*带权节点个数-1

哈夫曼树的构造过程

在未组成哈夫曼树的节点中寻找权值最小和第二小的节点，作为左右孩子,构造一个新节点作为父节点，父节点的权值为左右孩子的权值之和

哈夫曼树的表示

哈夫曼树用顺序存储的数组表示。

存储结构如下

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typedef struct TreeNode
{
    int weight;
    int parent;
    int lchild;
    int rchild;
} TreeNode;

typedef struct HaffTree
{
    struct TreeNode *node;
    int length;
} HaffTree;

哈夫曼树的初始化操作

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HaffTree *initTree(int *weight, int length)
{
    HaffTree *T = (HaffTree *)malloc(sizeof(HaffTree));
    T->node = (TreeNode *)malloc(sizeof(struct TreeNode) * (2 * length - 1));

    T->length = length;
    for (int i = 0; i < length; i++)
    {
        T->node[i].weight = weight[i];
        T->node[i].parent = -1;
        T->node[i].lchild = -1;
        T->node[i].rchild = -1;
    }
    return T;
}

上面我们已经说明哈夫曼树的节点树为带权节点个数*2-1，所以代码中我们申请了个数为为2 * length-1的哈夫曼结点数组

构建哈夫曼树

寻找最小节点和第二小节点。

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int *findMinWeight(HaffTree *T)
{
    int min = 10000;
    int secmin = 10000;
    int minidx = -1;
    int secidx = -1;

    for (int i = 0; i < T->length; i++)
    {
        if (T->node[i].parent == -1)
        {
           if (T->node[i].weight < min)
            {
                secmin = min;            // 次小值变成最小值
                secidx=minidx;

                min = T->node[i].weight; // 最小值更新
                minidx = i;             //最小值索引更新
            }
            else if (T->node[i].weight < secmin)
            {
                secmin = T->node[i].weight;
                secidx = i;
            }
        }
    }

    int *arr = (int *)malloc(sizeof(int) * 2);
    arr[0] = minidx;
    arr[1] = secidx;
    return arr;
}

构建哈夫曼树的具体过程

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void buildHaffTree(HaffTree *T)
{
    int length = T->length * 2 -1;

    for (int i = T->length; i < length; i++)
    {
        int *res = findMinWeight(T);
        int lchild = res[0], rchild = res[1];
        /*新节点*/
        T->node[i].weight = T->node[lchild].weight + T->node[rchild].weight;
        T->node[i].lchild = lchild;
        T->node[i].rchild = rchild;
        T->node[i].parent = -1;

        /*设置父节点*/
        T->node[lchild].parent = i;
        T->node[rchild].parent = i;

        T->length++;
    }

}

完整代码

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#include <stdio.h>
#include <stdlib.h>

#define MAXNODE 1024;
typedef struct TreeNode
{
    int weight;
    int parent;
    int lchild;
    int rchild;
} TreeNode;

typedef struct HaffTree
{
    struct TreeNode *node;
    int length;
} HaffTree;


HaffTree *initTree(int *weight, int length)
{
    HaffTree *T = (HaffTree *)malloc(sizeof(HaffTree));
    T->node = (TreeNode *)malloc(sizeof(struct TreeNode) * (2 * length - 1));

    T->length = length;
    for (int i = 0; i < length; i++)
    {
        T->node[i].weight = weight[i];
        T->node[i].parent = -1;
        T->node[i].lchild = -1;
        T->node[i].rchild = -1;
    }
    return T;
}

int *findMinWeight(HaffTree *T)
{
    int min = 10000;
    int secmin = 10000;
    int minidx = -1;
    int secidx = -1;

    for (int i = 0; i < T->length; i++)
    {
        if (T->node[i].parent == -1)
        {
           if (T->node[i].weight < min)
            {
                secmin = min;            // 次小值变成最小值
                secidx=minidx;

                min = T->node[i].weight; // 最小值更新
                minidx = i;             //最小值索引更新
            }
            else if (T->node[i].weight < secmin)
            {
                secmin = T->node[i].weight;
                secidx = i;
            }
        }
    }

    int *arr = (int *)malloc(sizeof(int) * 2);
    arr[0] = minidx;
    arr[1] = secidx;
    return arr;
}


void buildHaffTree(HaffTree *T)
{
    int length = T->length * 2 -1;

    for (int i = T->length; i < length; i++)
    {
        int *res = findMinWeight(T);
        int lchild = res[0], rchild = res[1];
        /*新节点*/
        T->node[i].weight = T->node[lchild].weight + T->node[rchild].weight;
        T->node[i].lchild = lchild;
        T->node[i].rchild = rchild;
        T->node[i].parent = -1;

        /*设置父节点*/
        T->node[lchild].parent = i;
        T->node[rchild].parent = i;

        T->length++;
    }

}
void preOrder(HaffTree *T, int idx)
{
    if (idx != -1){
        printf("%d->", T->node[idx].weight);
        preOrder(T, T->node[idx].lchild);
        preOrder(T, T->node[idx].rchild);
    }
}


int WPL(HaffTree *T){
    int wpl=0;
    for(int i=0;i<T->length-1;i++){
       if(T->node[i].lchild==-1&&T->node[i].rchild==-1){
            int j=i;
            int depth=0;
            while(T->node[j].parent!=0){
                j=T->node[j].parent;
                depth++;
            }
            wpl+=depth*T->node[i].weight;
       }
    }
    return wpl;
}
void secOrder(HaffTree *T, int idx)
{
    if (idx <= -1)
        return;
    preOrder(T, T->node[idx].lchild);
    printf("%d->", T->node[idx].weight);
    preOrder(T, T->node[idx].rchild);
}
int main()
{
    int weight[] = {5, 1, 3, 6, 11, 2, 4};
    HaffTree *T = initTree(weight, 7);
    buildHaffTree(T);
    preOrder(T, T->length - 1);
    printf("wpl=%d\n",WPL(T));
    return 0;
}