一、概念

给定n个权值作为n个叶子结点，构造一棵二叉树，若带权路径长度达到最小，称这样的二叉树为最优二叉树，也称为霍夫曼树(Huffman Tree)。

霍夫曼树

二、相关操作

构造霍夫曼树

三、实现

方法1:

构造霍夫曼树

参考代码：

#include <iostream>
#include <stdlib.h>

using namespace std;
const int MaxValue = 10000;//初始设定的权值最大值
const int MaxBit = 4;//初始设定的最大编码位数
const int MaxN = 10;//初始设定的最大结点个数

struct HaffNode//哈夫曼树的结点结构
{
    int weight;//权值
    int flag;//标记
    int parent;//双亲结点下标
    int leftChild;//左孩子下标
    int rightChild;//右孩子下标
};

struct Code//存放哈夫曼编码的数据元素结构
{
    int bit[MaxBit];//数组
    int start;//编码的起始下标
    int weight;//字符的权值
};

void Haffman(int weight[], int n, HaffNode haffTree[])
//建立叶结点个数为n权值为weight的哈夫曼树haffTree
{
    int j, m1, m2, x1, x2;
    //哈夫曼树haffTree初始化。n个叶结点的哈夫曼树共有2n-1个结点
    for (int i = 0; i<2 * n - 1; i++)
    {
        if (i<n)
            haffTree[i].weight = weight[i];
        else
            haffTree[i].weight = 0;
        //注意这里没打else那{}，故无论是n个叶子节点还是n-1个非叶子节点都会进行下面4步的初始化
        haffTree[i].parent = 0;
        haffTree[i].flag = 0;
        haffTree[i].leftChild = -1;
        haffTree[i].rightChild = -1;
    }
    //构造哈夫曼树haffTree的n-1个非叶结点
    for (int i = 0; i<n - 1; i++)
    {
        m1 = m2 = MaxValue;//Maxvalue=10000;(就是一个相当大的数)
        x1 = x2 = 0;//x1、x2是用来保存最小的两个值在数组对应的下标
 
        for (j = 0; j<n + i; j++)//循环找出所有权重中，最小的二个值--morgan
        {
            if (haffTree[j].weight<m1&&haffTree[j].flag == 0)
            {
                m2 = m1;
                x2 = x1;
                m1 = haffTree[j].weight;
                x1 = j;
            }
            else if(haffTree[j].weight<m2&&haffTree[j].flag == 0)
            {
                m2 = haffTree[j].weight;
                x2 = j;
            }
        }
        //将找出的两棵权值最小的子树合并为一棵子树
        haffTree[x1].parent = n + i;
        haffTree[x2].parent = n + i;
        haffTree[x1].flag = 1;
        haffTree[x2].flag = 1;
        haffTree[n + i].weight = haffTree[x1].weight + haffTree[x2].weight;
        haffTree[n + i].leftChild = x1;
        haffTree[n + i].rightChild = x2;
    }
}

void HaffmanCode(HaffNode haffTree[], int n, Code haffCode[])
//由n个结点的哈夫曼树haffTree构造哈夫曼编码haffCode
{
    Code *cd = new Code;
    int child, parent;
    //求n个叶结点的哈夫曼编码
    for (int i = 0; i<n; i++)
    {
        //cd->start=n-1;//不等长编码的最后一位为n-1,
        cd->start = 0;//,----修改从0开始计数--morgan
        cd->weight = haffTree[i].weight;//取得编码对应权值的字符
        child = i;
        parent = haffTree[child].parent;
        //由叶结点向上直到根结点
        while (parent != 0)
        {
            if (haffTree[parent].leftChild == child)
                cd->bit[cd->start] = 0;//左孩子结点编码0
            else
                cd->bit[cd->start] = 1;//右孩子结点编码1
                                      //cd->start--;
            cd->start++;//改成编码自增--morgan
            child = parent;
            parent = haffTree[child].parent;
        }
        //保存叶结点的编码和不等长编码的起始位
        //for(intj=cd->start+1;j<n;j++)
        for (int j = cd->start - 1; j >= 0; j--)//重新修改编码，从根节点开始计数--morgan
            haffCode[i].bit[cd->start - j - 1] = cd->bit[j];
 
        haffCode[i].start = cd->start;
        haffCode[i].weight = cd->weight;//保存编码对应的权值
    }
}

int main()
{
    int i, j, n = 4, m = 0;
    int weight[] = { 2,4,5,7 };
    HaffNode*myHaffTree = new HaffNode[2 * n - 1];
    Code*myHaffCode = new Code[n];
    if (n>MaxN)
    {
        cout << "定义的n越界，修改MaxN!" << endl;
        exit(0);
    }
    Haffman(weight, n, myHaffTree);
    HaffmanCode(myHaffTree, n, myHaffCode);
    //输出每个叶结点的哈夫曼编码
    for (i = 0; i<n; i++)
    {
        cout << "Weight=" << myHaffCode[i].weight << "  Code=";
        //for(j=myHaffCode[i].start+1;j<n;j++)
        for (j = 0; j<myHaffCode[i].start; j++)
            cout << myHaffCode[i].bit[j];
        m = m + myHaffCode[i].weight*myHaffCode[i].start;
        cout << endl;
    }
    cout << "huffman's WPL is:";
    cout << m;
    cout << endl;
    return 0;
}