哈夫曼树和哈夫曼编码

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，也称为哈夫曼树(Huffman tree)。

哈夫曼树的构造

 哈夫曼树的构造

假设有n个权值，则构造出的哈夫曼树有n个叶子结点。 n个权值分别设为 w1、w2、…、wn，则哈夫曼树的构造规则为：

(1) 将w1、w2、…，wn看成是有n 棵树的森林(每棵树仅有一个结点)；

(2) 在森林中选出两个根结点的权值最小的树合并，作为一棵新树的左、右子树，且新树的根结点权值为其左、右子树根结点权值之和；

(3)从森林中删除选取的两棵树，并将新树加入森林；

(4)重复(2)、(3)步，直到森林中只剩一棵树为止，该树即为所求得的哈夫曼树。

程序部分如下：

#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[MaxN]; //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;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;x1 = x2 = 0;for(j = 0; j < n+i;j++)//找出剩下的两个比较小的节点{if (haffTree[j].weight < m1 && haffTree[j].flag == 0){m2 = m1;x2 = x1;m1 = haffTree[j].weight;x1 = j;}elseif(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-1cd->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0elsecd->bit[cd->start] = 1;//右孩子结点编码1cd->start--;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(int j = cd->start+1; j < n; j++)haffCode[i].bit[j] = 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;int weight[] = {1,3,5,7};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++)cout << myHaffCode[i].bit[j];cout << endl;}system("pause");return 0;}