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)。哈夫曼树是带权路径长度最短的树，权值较大的结点离根较近。节点的带权长度是这样定义的：节点的权值*根节点到该节点的路径长度。树的带权路径长度(Weighted Path Length of Tree，简记为WPL)则是指所有节点的带权长度和。哈夫曼树就是使WPL最小的一种树，并且哈夫曼树是满二叉树。它的构造方法是哈夫曼方法。哈夫曼树是这样构造的：
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：(摘自百度百科)
提到哈夫曼树，这就不得不提到哈夫曼编码。哈夫曼编码(Huffman Coding)是一种编码方式，哈夫曼编码是可变字长编码(VLC)的一种。Huffman于1952年提出一种编码方法，该方法完全依据字符出现概率来构造异字头的平均长度最短的码字，有时称之为最佳编码，一般就叫做Huffman编码(有时也称为霍夫曼编码)。它的构造比较简单：先建好哈夫曼树，左子树的路径标记为0，右子树的路径标记为1。叶节点的哈夫曼编码就是根节点到叶节点的简单路径上的0、1序列。
下面给出构建哈夫曼树和哈夫曼编码的c++代码：
//Huffman树
#include<iostream>
#include<iomanip>
#include<stack>
using namespace std;
//最大权值
const int MAXWEIGHT = 100;
//Huffman节点
typedef struct
{
int weight; //节点权值
int parent; //父节点下标
int lchild; //左孩子下标
int rchild; //右孩子下标
}HuffNode, *HuffTree;
//HuffCode
typedef struct
{
//权值
int weight;
//用栈存储编码
stack<char> code;
}HuffCode;
/*
创建Huffman树
根据指定的权值数组创建Huffman树
*/
HuffNode *buildHuffTree(int weight[], int n)
{
if (!weight || n < 1)
return NULL;
/*
用顺序存储Huffman树
Huffman树是满二叉树
n为叶子节点数，则内部节点数是n-1，共有2*n-1个节点
*/
int m = 2 * n - 1; //节点总数
HuffNode* tree = new HuffNode[m];
//初始化Huffman树
int i, j;
for (i = 0; i < m; i++)
{
//把初始节点存储在数组前部
if (i < n)
tree[i].weight = weight[i];
tree[i].parent = tree[i].lchild = tree[i].rchild = -1;
}
/*
建树
只需n-1次循环
w1是当前最小权值，p1是其下标；w2是当前次最小权值，p2是其下标
*/
int w1, w2, p1, p2;
for (i = n; i < m; i++)
{
//每次循环前都得初始化权值和下标
w1 = w2 = MAXWEIGHT;
p1 = p2 = 0;
//寻找最小和次最小权值节点
for (j = 0; j < i; j++)
{
//父节点下标为-1，说明该节点未被使用
if (tree[j].parent == -1)
{
if (tree[j].weight < w1)
{
w2 = w1;
p2 = p1;
w1 = tree[j].weight;
p1 = j;
}
else if (tree[j].weight < w2)
{
w2 = tree[j].weight;
p2 = j;
}
}
}
//把产生的新节点放入位置i
tree[p1].parent = tree[p2].parent = i;
tree[i].weight = tree[p1].weight + tree[p2].weight;
tree[i].lchild = p1;
tree[i].rchild = p2;
}
return tree;
}
/*
根据Huffman树构建Huffman编码
*/
HuffCode* huffCode(HuffTree tree, int n)
{
HuffCode* Code = new HuffCode[n];
int i, child, parent;
for (i = 0; i < n; i++)
{
stack<char> stack;
child = i;
parent = tree[child].parent;
while (parent != -1)
{
//左孩子的分支是1
if (tree[parent].lchild == child)
stack.push('0');
else//右孩子的分支是1
stack.push('1');
child = parent;
parent = tree[child].parent;
}
Code[i].weight = tree[i].weight;
Code[i].code = stack;
}
return Code;
}
//打印Huffman树
void printHuffTree(HuffTree tree, int n)
{
int i;
cout.setf(ios::left);
for (i = 0; i < n; i++)
{
cout << setw(2) << i
<< " 权值:" << setw(3) << tree[i].weight
<< "父亲:" << setw(3) << tree[i].parent
<< "左孩子:" << setw(3) << tree[i].lchild
<< "右孩子:" << setw(3) << tree[i].rchild
<< endl;
}
}
int main()
{
cout << "***Huffman树***by David***" << endl;
int n;
cout << "输入权值个数 ";
while (cin >> n && n < 1)
cout << "出错，重新输入 ";
int *weight = new int[n];
cout << "输入权值序列 ";
int i = 0;
while (i < n)
cin >> weight[i++];
cout << "创建Huffman树" << endl;
HuffTree tree = buildHuffTree(weight, n);
printHuffTree(tree, 2 * n - 1);
cout << endl;
cout << "进行Huffman编码" << endl;
HuffCode *code = huffCode(tree, n);
for (i = 0; i < n; i++)
{
cout << "权值:" << setw(3) << code[i].weight << "Huffman编码:";
stack<char> stack = code[i].code;
while (!stack.empty())
{
cout << stack.top();
stack.pop();
}
cout << endl;
}
//释放空间
delete[]weight;
delete[]tree;
delete[]code;
system("pause");
return 0;
}
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专栏目录：数据结构与算法目录
原文地址：http://blog.csdn.net/zhangxiangdavaid/article/details/37696533
