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;
}