二叉树搜索树---AVL树插入节点

1.AVL树的概念：
AVL树或则是空树，或是具有下列性质的二叉搜索树

• 它的左右子树都是AVL树；
• 左子树和右子树的高度之差简称（平衡因子）的绝对值不超过1；

2.AVL树的实现原理
与二叉搜索树的节点插入方法相同，具体步骤如下：

当树不平衡时，我们需要做出旋转调整，有四种调整方法，分别为右单旋、左单旋、先左单旋再右单旋、先右单旋在左单旋

• 左单旋
  

• 右单旋
  

• 先右旋再左旋
  

• 先左旋再右旋
  

3.代码实现

#include<stdio.h>#include<iostream>using namespace std;template<class k,class v>struct AVLTreeNode{    AVLTreeNode(const k& key,const v& value)        :_key(key)        ,_value(value)        ,_bf(0)        ,_pLeft(NULL)        ,_pRight(NULL)        ,_pParent(NULL)    {}    AVLTreeNode<k,v>* _pLeft;    AVLTreeNode<k,v>* _pRight;    AVLTreeNode<k,v>* _pParent;    k _key;    v _value;    int _bf;};template<class k,class v>class AVLTree{public:    typedef AVLTreeNode<k,v> Node;    AVLTree()        :_pRoot(NULL)    {}    bool Insert(const k& key,const v& value)//插入节点    {        return _Insert(key,value);    }    bool IsBalanceTree()    {        return _IsBalanceTree(_pRoot);    }private:    bool _IsBalanceTree(Node* pRoot)//判断该树是否是平衡二叉树    {        if(pRoot == NULL)            return true;        size_t leftHeight = _Height(pRoot->_pLeft);        size_t rightHeight = _Height(pRoot->_pRight);        if(rightHeight-leftHeight != pRoot->_bf || abs(pRoot->_bf)>1)        {            cout<<pRoot->_key<<"-->"<<pRoot->_bf<<endl;            return false;        }        return _IsBalanceTree(pRoot->_pLeft)&&_IsBalanceTree(pRoot->_pRight);    }    size_t _Height(Node* pRoot)    {        if(pRoot == NULL)            return 0;        if(pRoot->_pLeft == NULL && pRoot->_pRight == NULL)            return 1;        size_t leftHeight = _Height(pRoot->_pLeft);        size_t rightHeight = _Height(pRoot->_pRight);        return 1+(leftHeight > rightHeight? leftHeight:rightHeight);    }    bool _Insert(const k& key,const v& value)    {        if(_pRoot == NULL)        {            _pRoot = new Node(key,value);            return true;        }        //找到要插入节点的位置        Node* pCur = _pRoot;        Node* pParent = NULL;        while(pCur)        {            if(key < pCur->_key)            {                pParent = pCur;                pCur = pCur->_pLeft;            }            else if(key > pCur->_key)            {                pParent = pCur;                pCur = pCur->_pRight;            }            else                return false;        }        //已经找到插入位置        pCur = new Node(key,value);        if(pParent->_key < key)        {            pParent->_pRight = pCur;            pCur->_pParent = pParent;        }        else        {            pParent->_pLeft = pCur;            pCur->_pParent = pParent;        }        while(pParent)        {            //更新平衡因子            if(pParent->_pLeft == pCur)                (pParent->_bf)--;            else                (pParent->_bf)++;            if(pParent->_bf == 0)//pParent只有左孩子或者只有右孩子，在其空指针域插入一个节点后，平衡因子为0，                return true;     //此时树的高度没有发生变化，该树任然是是AVL树            else if(pParent->_bf == -1 || pParent->_bf == 1)//pParent是叶子节点，插入节点后其平衡因子可能是1或-1            {                                             //此时树的高度可能发生了改变，我们要向上判断其祖先节点是否平衡                pCur = pParent;                pParent = pCur->_pParent;            }            else            {                //不满足平衡树，要做旋转处理                if(pParent->_bf == 2)//右子树                {                    if(pCur->_bf == 1)//右侧                        _RotateL(pParent);//左旋调整                    else//左侧                        _RotateRL(pParent);//先右旋再左旋                }                else//左子树                {                    if(pCur->_bf == -1)//左侧                        _RotateR(pParent);                    else//右侧                        _RotateLR(pParent);                }                break;//调整完后跳出循环            }        }        return true;    }    void InOrder()    {        cout<<"InOrder ";        _InOrder(_pRoot);        cout<<endl;    }private:    void _InOrder(Node*& pRoot)    {        if(pRoot)        {            _InOrder(pRoot->_pLeft);            cout<<pRoot->_key<<" ";            _InOrder(pRoot->_pRight);        }    }    void _RotateL(Node* pParent)    {        Node* pSubR = pParent->_pRight;        Node* pSubRL =pSubR->_pLeft;        pParent->_pRight = pSubRL;        if(pSubRL)//情况四之一            pSubRL->_pParent = pParent;        pSubR->_pLeft = pParent;        Node* pPParent = pParent->_pParent;        pParent->_pParent = pSubR;        if(pPParent == NULL)        {            _pRoot = pSubR;            pSubR->_pParent = NULL;        }        else if(pPParent->_pLeft == pParent)        {            pPParent->_pLeft = pSubR;            pSubR->_pParent = pPParent;        }        else        {            pPParent->_pRight = pSubR;            pSubR->_pParent = pPParent;        }        //更新平衡因子        pParent->_bf = pSubR->_bf = 0;    }    void _RotateR(Node* pParent)    {        Node* pSubL = pParent->_pLeft;        Node* pSubLR = pParent->_pRight;        pParent->_pLeft = pSubLR;        if(pSubLR)            pSubLR->_pParent = pParent;        pSubL->_pRight = pParent;        Node* pPParent = pParent->_pParent;        pParent->_pParent = pSubL;        if(pPParent == NULL)        {            _pRoot = pSubL;            pSubL->_pParent = NULL;        }        else if(pPParent->_pLeft == pParent)        {            pPParent->_pLeft = pSubL;            pSubL->_pParent = pPParent;        }        else        {            pPParent->_pRight = pSubL;            pSubL->_pParent = pPParent;        }        //更新平衡因子        pParent->_bf = pSubL->_bf = 0;    }    void _RotateRL(Node* pParent)//先右单旋再左单旋    {        Node* pSubR = pParent->_pRight;        Node* pSubRL = pSubR->_pLeft;        int bf = pSubRL->_bf;        _RotateR(pParent->_pRight);        _RotateL(pParent);        //判断平衡因子应该如何更新        if(bf == 1)            pParent->_bf = -1;        else            pSubR->_bf = 1;    }    void _RotateLR(Node* pParent)//先左单旋再右单旋    {        Node* pSubL = pParent->_pLeft;        Node* pSubLR = pSubL->_pRight;        int bf = pSubLR->_bf;        _RotateL(pParent->_pLeft);        _RotateR(pParent);        //判断平衡因子应该如何修改        if(bf == 1)            pSubL->_bf = -1;        else            pParent->_bf = 1;    }private:    Node* _pRoot;};

测试代码：

void funtest(){    int array[] = {2,1,4,3,5,6};//测试左单旋    AVLTree<int,int> bst;    for(size_t idx= 0 ;idx < sizeof(array)/sizeof(array[0]);++idx)        bst.Insert(array[idx],idx);    AVLTree<int,int> bst1;    int array1[] = {5,3,6,2,1,4};//测试右单旋    for(size_t idx1= 0 ;idx1 < sizeof(array1)/sizeof(array1[0]);++idx1)        bst1.Insert(array1[idx1],idx1);}void funtest1(){    int array[] = {3,2,1,4,5,6,7,10,9,8};//测试左单旋    AVLTree<int,int> bst;    for(size_t idx= 0 ;idx < sizeof(array)/sizeof(array[0]);++idx)        bst.Insert(array[idx],idx);    if(bst.IsBalanceTree())    {        cout<<"IS AVLTree"<<endl;    }    else    {        cout<<"NO IS AVLTree"<<endl;    }}int main(){    //funtest();    funtest1();    getchar();    return 0;}