次优二叉树

　　在有序序列的查找中，如果各个元素的查找概率都是一样的，那么二分查找是最快的查找算法，但是如果查找元素的查找概率是不一样的，那么用二分查找就不一定是最快的查找方法了，可以通过计算ASL来得知。所以基于这种查找元素概率不想等的有序序列，可以通过构造最优二叉树的方法，使得该二叉树的带权路径长度最小，这样的二叉树的构造代价是非常大的，所以用一种近似的算法，构造次优查找树，该树的带权路径长度近似达到最小。

　　数据结构中算法描述为：

　　

#include <iostream>
#include <cmath>
#include <cstdlib>
#include <iomanip>

using namespace std;

typedef struct treenode
{
    char data;
    int weight;
    treenode *left;
    treenode *right;
}Treenode,*Treep;

//初始化二叉树
void init_tree(Treep &root)
{
    root=NULL;
    cout<<"初始化成功!"<<endl;
}

//创建二叉树
void SecondOptimal(Treep &rt, char R[],int sw[], int low, int high)
{
    //由有序表R[low....high]及其累积权值表sw(其中sw[0]==0)递归构造次优查找树T
    int i = low;
    //int min = abs(sw[high] - sw[low]);
    int dw = sw[high] - sw[low-1];
    int min = dw;
    for(int j=low+1; j<=high; ++j)        //选择最小的ΔPi值
    {
        if(abs(dw-sw[j]-sw[j-1]) < min)
        {
            i=j;
            min=abs(dw-sw[j]-sw[j-1]);
        }
    }
    rt=new Treenode;
    rt->data=R[i];        //生成节点
    if(i==low)            //左子树为空
        rt->left = NULL;
    else                //构造左子树
        SecondOptimal(rt->left, R, sw, low, i-1);

    if(i==high)            //右子树为空
        rt->right = NULL;
    else                //构造右子树
        SecondOptimal(rt->right, R, sw, i+1, high);
}//SecondOptimal

//前序遍历二叉树
void pre_order(Treep rt)
{
    if(rt)
    {
        cout<<rt->data<<"  ";
        pre_order(rt->left);
        pre_order(rt->right);
    }
}

//中序遍历二叉树
void in_order(Treep rt)
{
    if(rt)
    {
        in_order(rt->left);
        cout<<rt->data<<"  ";
        in_order(rt->right);
    }
}

//后序遍历二叉树
void post_order(Treep rt)
{
    if(rt)
    {
        post_order(rt->left);
        post_order(rt->right);
        cout<<rt->data<<"  ";
    }
}

//查找二叉树中是否存在某元素
int seach_tree(Treep &rt,char key)
{
    if(rt==NULL) 
        return 0; 
    else 
    { 
        if(rt->data==key) 
        {
            return 1;
        }
        else
        {
            if(seach_tree(rt->left,key) || seach_tree(rt->right,key))
                return 1;    //如果左右子树有一个搜索到，就返回1
            else
                return 0;    //如果左右子树都没有搜索到，返回0
        }
    }
}

int main()
{
    Treep root;
    init_tree(root);        //初始化树
    int low=1, high=10;
    int *weight, *sw;
    char *R;
    
    R=new char[high];
    for(int i=low; i<high; i++)
        R[i]='A'+i-1;
    cout<<"构造次优查找树的点R[]:"<<endl;
    for(int i=low; i<high; i++)
        cout<<setw(3)<<R[i]<<"  ";
    cout<<endl;
    
    weight=new int[high];
    weight[0]=0;
    weight[1]=1;
    weight[2]=1;
    weight[3]=2;
    weight[4]=5;
    weight[5]=3;
    weight[6]=4;
    weight[7]=4;
    weight[8]=3;
    weight[9]=5;
    cout<<"构造次优查找树的点的权值weight[]:"<<endl;
    for(int i=low; i<high; i++)
        cout<<setw(3)<<weight[i]<<"  ";
    cout<<endl;
        
    sw=new int[high];
    sw[0]=0;
    for(int i=low; i<high; i++)
    {
        sw[i]=sw[i-1]+weight[i];
    }
    cout<<"构造次优查找树的点累积权值sw[]:"<<endl;
    for(int i=low; i<high; i++)
        cout<<setw(3)<<sw[i]<<"  ";
    cout<<endl;
        
    //创建二叉树
    SecondOptimal(root, R, sw, low, high-1);
    
    cout<<"前序遍历序列是:"<<endl;
    pre_order(root);
    cout<<endl;
    
    cout<<"中序遍历序列是:"<<endl;
    in_order(root);
    cout<<endl;

    cout<<"后序遍历序列是:"<<endl;
    post_order(root);
    cout<<endl;

    //查找二叉树中是否存在某元素
    cout<<"输入要查找的元素!"<<endl;
    char ch;
    cin>>ch;
    if(seach_tree(root,ch)==1)
        cout<<"Yes"<<endl;
    else
        cout<<"No"<<endl;
    while(1);
    return 0;
}

　　运行结果如下：