题目：http://pta.patest.cn/pta/test/16/exam/4/question/668

PTA – Data Structures and Algorithms (English) – 5-6

An AVL tree is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Figures 1-4 illustrate the rotation rules.

LL：RR：

RL：                                  

LR：                         

Now given a sequence of insertions, you are supposed to tell the root of the resulting AVL tree.

Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer N (≤20) which is the total number of keys to be inserted. Then N distinct integer keys are given in the next line. All the numbers in a line are separated by a space.

Output Specification:

For each test case, print the root of the resulting AVL tree in one line.

Sample Input 1:

5
88 70 61 96 120

Sample Output 1:

70

Sample Input 2:

7
88 70 61 96 120 90 65

Sample Output 2:

88

分析：

1. 树的结点结构

typedef struct node
{
int data;
node* left;
node* right;
int height;
}AVLTreeNode,*AVLTree;

2. 函数声明

int GetHeight(AVLTree A) //获取当前树高
int Max(int x,int y)     //用于更新树高
//以下操作返回调整后的AVL树
AVLTree SingleL_Rotation(AVLTree A)    //左单旋：LL
AVLTree SingleR_Rotation(AVLTree A)    //右单旋：RR
AVLTree DoubleLR_Rotation(AVLTree A)   //右左双旋：RL
AVLTree DoubleRL_Rotation(AVLTree A)   //左右双旋：LR
AVLTree AVL_Insertion(int x,AVLTree T) //将x插入AVL树T中

3. 函数实现 (以左旋为例)：

//左单旋:LL
AVLTree SingleL_Rotation(AVLTree A)
{
    //!注：A必须有一个左子节点B
    //!左单旋后，更新A和B的高度，返回新的根节点

    AVLTree 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),A->height)+1;
    return B;
}
//左右双旋;LR
AVLTree DoubleLR_Rotation(AVLTree A)
{
    //!注：A必须有一个左子结点B,且B必须有一个右子节点C
    //!做两次单旋，返回新的根节点：C

    A->left=SingleR_Rotation(A->left); //!B和C做右单旋，返回C
    return SingleL_Rotation(A); //!A和做左单旋，返回C
}

完整代码：

#include <iostream>
using namespace std;

typedef struct node
{
    int data;
    node* left;
    node* right;
    int height;
}AVLTreeNode,*AVLTree;

int GetHeight(AVLTree A)
{
    if(A==NULL)return -1;
    return A->height;
}
int Max(int x,int y)
{
   return (x>y)?x:y;
}

//!左单旋:LL
AVLTree SingleL_Rotation(AVLTree A)
{
    //!注：A必须有一个左子节点B
    //!左单旋后，更新A和B的高度，返回新的根节点

    AVLTree 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),A->height)+1;
    return B;
}
//!右单旋:RR
AVLTree SingleR_Rotation(AVLTree A)
{
    AVLTree C=A->right;
    A->right=C->left;
    C->left=A;
    A->height=Max(GetHeight(A->left),GetHeight(A->right))+1;
    C->height=Max(A->height,GetHeight(C->right))+1;
    return C;
}
//!左右双旋;LR
AVLTree DoubleLR_Rotation(AVLTree A)
{
    //!注：A必须有一个左子结点B,且B必须有一个右子节点C
    //!做两次单旋，返回新的根节点：C

    A->left=SingleR_Rotation(A->left); //!B和C做右单旋，返回C
    return SingleL_Rotation(A); //!A和做左单旋，返回C
}
//!右左双旋:RL
AVLTree DoubleRL_Rotation(AVLTree A)
{
    A->right=SingleL_Rotation(A->right);
    return SingleR_Rotation(A);
}

//!将x插入AVL树T中，并且返回调整后的AVL树
AVLTree AVL_Insertion(int x,AVLTree T)
{
    if(!T)  //!若插入空树，则新建包含一个节点的树
    {
        T=new AVLTreeNode;
        T->data=x;
        T->height=0;
        T->left=T->right=NULL;
    }
    else if(x<T->data)  //!插入T的左子树
    {
        T->left=AVL_Insertion(x,T->left);
        if(GetHeight(T->left)-GetHeight(T->right)==2)
        {
            //!需左旋
            if(x<T->left->data)
                T=SingleL_Rotation(T);  //!左单旋:LL
            else
                T=DoubleLR_Rotation(T); //!左右双旋:LR
        }
    }
    else if(x>T->data)  //!插入T的右子树
    {
        T->right=AVL_Insertion(x,T->right);
        if(GetHeight(T->left)-GetHeight(T->right)==-2)
        {
            //!需右旋
            if(x>T->right->data)
                T=SingleR_Rotation(T);  //!右单旋:RR
            else
                T=DoubleRL_Rotation(T); //!右左双旋:RL
        }
    }
    else    //! x==T->data, 无需插入
        return T;

    //!更新树高
    T->height=Max(GetHeight(T->left),GetHeight(T->right))+1;
    return T;
}

int main()
{
    int n,x;
    cin >> n;
    AVLTree root=NULL;
    for(int i=0;i<n;i++)
    {
        cin >> x;
        root=AVL_Insertion(x,root);
    }
    cout << root->data << endl;
    return 0;
}