数据结构(7) -- 平衡二叉树

此版本是在数据结构（6）的基础上改造的。实现了构建平衡二叉树功能。

//BinTree.h

#ifndef BINTREE_H_

#define BINTREE_H_

#define ElemType int

typedef struct _PNode

{

    ElemType data;

    _PNode *left;

    _PNode *right;

    int height;

}PNode;



class BinTree

{

private:

    PNode *root;

public:

    BinTree();

    ~BinTree();

    PNode* GetRoot();    //获得根节点

    void SetRoot(PNode *p) { root = p; }    //设置根节点

    PNode* Find(ElemType x , PNode *p);        //查找元素x所在位置

    PNode* FindMin(PNode *p);    //查找二叉搜索树的最小值

    PNode* FindMax(PNode *p);    //查找二叉搜索树的最大值

    PNode* Insert(ElemType x, PNode *p);  //构建二叉搜索树，所插入的元素按二叉搜索树排列

    PNode* Delete(ElemType x, PNode *p);  //删除二叉搜索树的元素    

    //遍历二叉树

    void PreOrder(PNode *p);  //先序遍历

    void CenOrder(PNode *p);  //中序遍历

    void Trave(PNode *p);     //层次遍历

    int GetHeight(PNode *p);

    PNode* AVL_Insertion(ElemType x, PNode *p);

    PNode* SingleLeftRotation(PNode *p); //左单旋

    PNode* DoubleLeftRightRotation(PNode *p); //左右双旋

    PNode* SingleRightRotation(PNode *p); //右单旋

    PNode* DoubleRightLeftRotation(PNode *p); //右左双旋

    int Max(int a, int b); //求最大值

};

#endif





/////////////////////////////////////////////////////////////////////

//BinTree.cpp

#include "BinTree.h"

#include <iostream>

#include <queue>



BinTree::BinTree()

{

    root = NULL;

}



BinTree::~BinTree(){}



PNode* BinTree::GetRoot()

{

    return root;

}

PNode* BinTree::Find(ElemType x, PNode *p)

{

    /*

    //尾递归实现

    if (p == NULL)

        return NULL;

    if (x > p->data)

        return Find(x, p->right);

    else if (x < p->data)

        return Find(x, p->left);

    else

        return p;*/

    while (p != NULL)

    {

        if (x > p->data)

            p = p->right;

        else if (x < p->data)

            p = p->left;

        else

            return p;

    }

    return NULL;

}



PNode* BinTree::FindMin(PNode *p)

{

    //递归查找

    if (p == NULL)

        return NULL;

    else if (p->left == NULL)

        return p;

    else

        return FindMin(p->left);

}



PNode* BinTree::FindMax(PNode *p)

{

    if (p != NULL)

        while (p->right != NULL)

            p = p->right;

    return p;

}



//遍历二叉树

void BinTree::PreOrder(PNode *p)

{

    if (p == NULL)

        return;

    std::cout << p->data << " ";

    PreOrder(p->left);

    PreOrder(p->right);

}



void BinTree::CenOrder(PNode *p)

{

    if (p == NULL)

        return;

    CenOrder(p->left);

    std::cout << p->data << " ";

    CenOrder(p->right);

}



PNode* BinTree::Insert(ElemType x, PNode *p)

{

    if (p == NULL)

    {

        p = new PNode();

        p->data = x;

        p->left = p->right = NULL;

    }

    else

    {

        if (x < p->data)

            p->left = Insert(x, p->left);

        else if (x > p->data)

            p->right = Insert(x, p->right);

    }

    return p;

}



PNode* BinTree::Delete(ElemType x, PNode *p)

{

    PNode *tmp;

    if (p == NULL)

        std::cout << "要删除的元素未找到" << std::endl;

    else if (x < p->data)  //查找要删除的节点

        p->left = Delete(x, p->left);

    else if (x > p->data)

        p->right = Delete(x, p->right);

    else

    {

        //删除的元素左右节点都在,删除右子树的最小节点

        if (p->left != NULL && p->right != NULL)

        {

            tmp = FindMin(p->right);  //查找右子树的最小节点

            p->data = tmp->data;          //将右字数的最小值与当前节点的值互换    

            p->right = Delete(p->data, p->right);    //删除右字数最小节点

        }

        else

        {//删除的元素只有左节点，或只有右节点，或是叶子节点

            tmp = p;

            if (p->left == NULL)  //只有右节点，直接用右节点替换要删除节点

                p = p->right;

            else if(p->right == NULL) //只有左节点，直接用左节点替换要删除节点

                p = p->left;

            delete tmp;    //如果删除节点是叶子节点，直接删除当前节点

        }        

    }

    return p;

}



void BinTree::Trave(PNode *p)    //层次遍历

{

    std::queue<PNode*> q;

    q.push(p);

    while (!q.empty())

    {

        PNode *s = q.front();

        std::cout << s->data << " ";

        if (s->left != NULL)

            q.push(s->left);

        if (s->right != NULL)

            q.push(s->right);

        q.pop();

    }

}



int BinTree::GetHeight(PNode *p)

{

    int hl;

    int hr;

    int maxh;

    if (p)

    {

        hl = GetHeight(p->left);

        hr = GetHeight(p->right);

        maxh = (hl > hr) ? hl : hr;

        return (maxh + 1);

    }

    else

        return -1;

}



PNode* BinTree::AVL_Insertion(ElemType x, PNode *p)

{

    //将x插入AVL树T中，并且返回调整后的AVL树

    if (p == NULL)    //如果插入空树，则新建包含一个节点的树

    {

        p = new PNode();

        p->data = x;

        p->height = 0;

        p->left = p->right = NULL;

    }

    else if (x < p->data) //插入T的左子树

    {

        p->left = AVL_Insertion(x, p->left);

        if (GetHeight(p->left) - GetHeight(p->right) == 2)

            if (x < p->left->data) //需要左旋

                p = SingleLeftRotation(p); //左单旋

            else

                p = DoubleLeftRightRotation(p); //左右双旋

    } //插入左子树结束

    else if (x > p->data)

    {

        p->right = AVL_Insertion(x, p->right);

        if (GetHeight(p->right) - GetHeight(p->left) == 2)

            if (x > p->right->data) //需要又旋

                p = SingleRightRotation(p); //右单旋

            else

                p = DoubleRightLeftRotation(p); //右左双旋

    }

    //x == p->data无需插入

    p->height = Max(GetHeight(p->left), GetHeight(p->right)) + 1;

    return p;

}



int BinTree::Max(int a, int b)

{

    return (a > b ? a : b);

}



PNode* BinTree::SingleLeftRotation(PNode *A) //左单旋

{

    //注意，A必须有一个左子结点

    PNode *B = A->left;

    A->left = B->right;

    B->right = A;

    A->height = Max(GetHeight(A->left), GetHeight(A->right)) + 1;

    B->height = Max(GetHeight(B->left), GetHeight(A->right)) + 1;

    return B;

}



PNode* BinTree::DoubleLeftRightRotation(PNode *A) //左右双旋

{

    A->left = SingleRightRotation(A->left);

    return SingleLeftRotation(A);

}



PNode* BinTree::SingleRightRotation(PNode *A) //右单旋

{

    PNode *B = A->right;

    A->right = B->left;

    B->left = A;

    A->height = Max(GetHeight(A->left), GetHeight(A->right)) + 1;

    B->height = Max(GetHeight(B->left), GetHeight(A->right)) + 1;

    return B;

}



PNode* BinTree::DoubleRightLeftRotation(PNode *A) //右左双旋

{

    A->right = SingleLeftRotation(A->right);

    return SingleRightRotation(A);

}





////////////////////////////////////////////////////////

//测试

#include <iostream>

#include "BinTree.h"

using namespace std;



int main()

{

    BinTree bt;

    //构造二分查找树

    bt.SetRoot(bt.AVL_Insertion(2, bt.GetRoot())); //注意跟新根节点    

    bt.SetRoot(bt.AVL_Insertion(3, bt.GetRoot()));

    bt.SetRoot(bt.AVL_Insertion(4, bt.GetRoot()));

    bt.SetRoot(bt.AVL_Insertion(5, bt.GetRoot()));

    bt.SetRoot(bt.AVL_Insertion(6, bt.GetRoot()));

    bt.SetRoot(bt.AVL_Insertion(8, bt.GetRoot()));

    bt.SetRoot(bt.AVL_Insertion(2, bt.GetRoot()));

    bt.SetRoot(bt.AVL_Insertion(1, bt.GetRoot()));

    cout << bt.GetHeight(bt.GetRoot()) << endl;



    cout << "先序遍历: " << endl;

    bt.PreOrder(bt.GetRoot());

    cout << endl;

    cout << "中序遍历: " << endl;

    bt.CenOrder(bt.GetRoot());

    cout << endl;

    cout << "层次遍历：" << endl;

     bt.Trave(bt.GetRoot());

    cout << endl;

    cout << "最小值:" << bt.FindMin(bt.GetRoot())->data << endl;

    cout << "最大值:" << bt.FindMax(bt.GetRoot())->data << endl;

    ///////////////////////////////////////////////////////////////////

    //cout << endl << "删除节点 5 " << endl;

    //bt.Delete(4, bt.GetRoot());

    cout << "先序遍历: " << endl;

    bt.PreOrder(bt.GetRoot());

    cout << endl;

    cout << "中序遍历: " << endl;

    bt.CenOrder(bt.GetRoot());

    cout << endl;

    cout << "层次遍历：" << endl;

    bt.Trave(bt.GetRoot());

    cout << endl;

    cout << "最小值:" << bt.FindMin(bt.GetRoot())->data << endl;

    cout << "最大值:" << bt.FindMax(bt.GetRoot())->data << endl;

    system("pause");

    return 0;

}