链表的访问速度太慢，不适合大量的输入数据。而树的大部分运行时间平均为O（logN）。

定义树的一种自然的额方式是递归的方法。

1.实现二叉树

TianryTree.h


typedef int ElementType;

#ifndef BINARYTREE_H_INCLUDED
#define BINARYTREE_H_INCLUDED


//二叉树声明

struct TreeNode;
typedef struct TreeNode *PtrToNode;
typedef PtrToNode pNode;
typedef PtrToNode Tree;

Tree MakeEmpty( Tree &T);
pNode Find(ElementType X, Tree T);
pNode FindMin(Tree T);
pNode FindMax(Tree T);
void TraverTree(Tree T);

Tree Insert(ElementType X, Tree T);
Tree Delete(ElementType X, Tree T);

ElementType Retrieve(pNode P);



#endif // BINARYTREE_H_INCLUDED

BinaryTree.cpp

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

#include "BinaryTree.h"
#include "fatal.h"

using namespace std;



//二叉树节点声明
struct TreeNode
{
    ElementType Element;
    Tree Left;
    Tree Right;
};

//空树
Tree MakeEmpty(Tree &T)
{
    if(T != NULL)
    {
        MakeEmpty(T->Left);
        MakeEmpty(T->Right);
        free(T);
        T=NULL;
    }

    cout << "the empty operation is end !!! " << endl << endl;
    return NULL;

}

//二叉查找数的find操作

pNode Find(ElementType X, Tree T)
{
    if(T==NULL)
    {
        cout << "NOT FIND The ELEMENT "<< X <<  endl << endl;
        return NULL;
    }
    if(X < T->Element)
    {
        cout << "serching " << T->Element << " Left SubTree..." << endl;
        return Find(X, T->Left);
    }
    else if(X > T->Element)
    {
        cout << "searching " << T->Element << " Right SubTree..."<< endl;
        return Find(X, T->Right);
    }
    else
    {
        cout << "the Element " << X << " is find !!! " << endl << endl;
        return T;
    }

}

//FinMin实现

pNode FindMin(Tree T)
{
    if(T==NULL)
    {
        return NULL;
    }
    else if(T->Left == NULL)
    {
        cout << "the min is : " << T->Element << endl << endl;
        return T;
    }
    else
        return FindMin(T->Left);
}


pNode FindMax(Tree T)
{
    if(T != NULL)
    {
        while(T->Right != NULL)
        {
            T=T->Right;
        }
    }
    cout << "the max is : " << T->Element << endl << endl;
    return T;
}


Tree Insert(ElementType X, Tree T)
{
    if(T == NULL)
    {
        T = (struct TreeNode *)malloc(sizeof(struct TreeNode));
        if(T == NULL)
        {
            FatalError("out of space!!!");
        }
        else
        {
            T->Element = X;
            T->Left = T->Right = NULL;
            cout << T->Element <<" Element! "<< endl;
        }
    }
    else if(X < T->Element)
    {
        T->Left = Insert(X, T->Left);
    }
    else if(X > T->Element)
    {
        T->Right = Insert(X, T->Right);
    }
    cout << "insert is OK !!!" << endl << endl;
    return T;
}


void TraverTree(Tree T)
{
    if(T == NULL)
    {
       cout << "Travel --- the Tree is empty!!!" << endl << endl;
    }
    else
    {
        cout << "node " << T->Element << endl << endl;

        TraverTree(T->Left);
        TraverTree(T->Right);
    }
}


int main()
{
    cout << "The Program is Starting ...." << endl << endl;
    Tree T = NULL;

    T = Insert(5,T);
    T = Insert(2,T);
    T = Insert(1,T);
    T = Insert(3,T);
    T = Insert(7,T);
    T = Insert(6,T);
    T = Insert(9,T);
    Find(1,T);
    FindMin(T);
    FindMax(T);
    TraverTree(T);
    MakeEmpty(T);
    TraverTree(T);


    return 0;
}

2.AVLTree,带有平衡条件的二叉查找树。树的深度是O（logN）。每个节点都必须有相同高度的左子树和右子树。

AVLTree.h


typedef int ElementType;

#ifndef AVLTREE_H_INCLUDED
#define AVLTREE_H_INCLUDED




struct AvlNode;
typedef struct AvlNode *PtrToNode;

typedef PtrToNode pNode;
typedef PtrToNode AvlTree;

AvlTree MakeEmpty(AvlTree &T);
pNode Find(ElementType X, AvlTree T);
pNode FindMin(AvlTree T);
pNode FindMax(AvlTree T);
AvlTree Insert(ElementType X, AvlTree T);
AvlTree Delete(ElementType X, AvlTree T);
ElementType Retrieve(pNode P);
static pNode SingleRotateWithLeft(pNode K);
static pNode DoubleRotateWithLeft(pNode K);
static pNode SingleRotateWithRight(pNode K);
static pNode DoubleRotateWithRight(pNode K);
void TraverTree(AvlTree T);


#endif // AVLTREE_H_INCLUDED

AVLTree.cpp


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

#include "AVLTree.h"
#include "fatal.h"

using namespace std;


//AVL节点声明
struct AvlNode
{
    ElementType Element;
    AvlTree Left;
    AvlTree Right;
    int Height;
};

//取两个数中较大的数
int Max(int a, int b)
{
    if(a<b)
    {
        return b;
    }
    else
    {
        return a;
    }
}

//计算节点高度函数,空树高度为0
static int Height(pNode P)
{
    if(P == NULL)
    {
        return 0;
    }
    else
        return P->Height;
}

AvlTree Insert(ElementType X, AvlTree T)
{
    if(T == NULL)
    {
        T = (struct AvlNode *)malloc(sizeof(struct AvlNode));
        if(T == NULL)
        {
            FatalError("out of space !!!");
        }
        else
        {
            T->Element = X;
            T->Height = 0;
            T->Left = NULL;
            T->Right = NULL;
            cout << "insert the element " << T->Element << " in AvlTree ! " << T->Height << endl;
        }
    }
    else
    {
        if(X<T->Element)
        {
            T->Left = Insert(X, T->Left);
            if(Height(T->Left) - Height(T->Right) ==2)
            {
                if(X<T->Left->Element)
                {
                    T = SingleRotateWithLeft(T);
                }
                else
                {
                    T = DoubleRotateWithLeft(T);
                }
            }
        }
        if(X>T->Element)
        {
            T->Right = Insert(X, T->Right);
            if(Height(T->Right) - Height(T->Left) ==2)
            {
                if(X>T->Right->Element)
                {
                   T = SingleRotateWithRight(T);
                }
                else
                {
                    T = DoubleRotateWithRight(T);
                }
            }

        }
    }
    T->Height = Max(Height(T->Left), Height(T->Right)) + 1;
    return T;
}

static pNode SingleRotateWithLeft(pNode K2)
{
    pNode K1;
    K1 = K2->Left;
    K2->Left = K1->Right;
    K1->Right = K2;

    K2->Height = Max(Height(K2->Left), Height(K2->Right)) +1 ;
    K1->Height = Max(Height(K1->Left), K2->Height) +1;
    return K1;
}
static pNode SingleRotateWithRight(pNode K1)
{
    pNode K2;
    K2 = K1->Right;
    K1->Right = K2->Left;
    K2->Left = K1;

    K2->Height = Max(Height(K1), Height(K2->Right)) + 1;
    K1->Height = Max(Height(K1->Left), Height(K1->Right))+ 1;

    return K2;
}
static pNode DoubleRotateWithLeft(pNode K3)
{
    K3->Left = SingleRotateWithRight(K3->Left);
    return SingleRotateWithLeft(K3);
}
static pNode DoubleRotateWithRight(pNode K3)
{
    K3->Right = SingleRotateWithLeft(K3->Right);
    return SingleRotateWithRight(K3);
}

void TraverTree(AvlTree T)
{
    if(T == NULL)
    {
       cout << "Travel --- the Tree is empty!!!" << endl << endl;
    }
    else
    {
        cout << "Node: " << T->Element << " Height: " << T->Height << endl << endl;

        TraverTree(T->Left);
        TraverTree(T->Right);
    }
}


int main()
{
    cout << "AVLTree program is starting ..." << endl << endl << endl;
    AvlTree T = NULL;
    cout << "insert element ..." << endl << endl;
    T = Insert(3,T);
    T = Insert(2,T);
    T = Insert(1,T);
    T = Insert(10,T);
    T = Insert(13,T);
    cout << "print the node ..." << endl << endl;
    TraverTree(T);

    return 0;
}

#include

ifstream fin;

fin.open("1.txt");

int a[10];

fin >> a[i];

fin.close();

ofstream fout;