the Root Of AVL

题目来源：陈越《数据结构》配套PTA;

04-树5 Root of AVL Tree   (25分)

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.

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 $N$ ($\le 20$) 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:

588 70 61 96 120

Sample Output 1:

70

Sample Input 2:

788 70 61 96 120 90 65

Sample Output 2:

88

解决思路：用课本提供的大部分程序，实现AVL插入和旋转，左旋，右旋，左-右旋，右-左旋；最后提取出根节点的数据：

代码如下：

#include <stdio.h>  #include <stdlib.h>  /*Name: Root of AVLCopyright: Author: demossesDate: 23/04/17 11:11Description: */typedef  int ElementType;typedef struct AVLNode *Position; typedef Position AVLTree;struct AVLNode {      ElementType Data;      ElementType Height;      AVLTree Left;      AVLTree Right;  };    AVLTree Insert(int X, AVLTree T);  int GetHeight(Position T);  int Max(int a, int b);  AVLTree SingleLeft(Position K);  AVLTree SingleRight(Position K);  AVLTree DoubleLeftRight(Position K);  AVLTree DoubleRightLeft(Position K);    int main(void) {      AVLTree T = NULL;      int N;      scanf("%d", &N);      while (N--) {          int x;          scanf("%d", &x);          T=Insert(x, T);      }      if (T)          printf("%d", T->Data);      return 0;  }    AVLTree Insert(ElementType X, AVLTree T) {      if (!T) {  /*插入为空树*/         T = (AVLTree)malloc(sizeof(struct AVLNode));          T->Data = X;          T->Height = 1;          T->Left = T->Right = NULL;      }      else if (X < T->Data) {       /*插入T的左子树*/         T->Left = Insert(X, T->Left);           /*需要旋转*/         if (GetHeight(T->Left) - GetHeight(T->Right) == 2) {              if (X < T->Left->Data)   /*左单旋*/                 T = SingleLeft(T);              else               /*左-右双旋*/                 T = DoubleLeftRight(T);          }      }      else if (X > T->Data) {            /*插入T的右子树*/        T->Right = Insert(X, T->Right);     /*需要旋转*/         if (GetHeight(T->Right) - GetHeight(T->Left) == 2) {              if (X > T->Right->Data)                    /*右单旋*/                 T = SingleRight(T);              else                /*左-右双旋*/                T = DoubleRightLeft(T);          }      }      T->Height = Max(GetHeight(T->Left), GetHeight(T->Right)) + 1;      return T;  }    int GetHeight(AVLTree T) {     int HL,HR,MaxH;      if(T){   HL=GetHeight(T->Left);    HR=GetHeight(T->Right);   MaxH=Max(HL,HR);   return (MaxH+1);}  else     return 0;}    int Max(int a,int b) {      return (a > b) ? a : b;  }    AVLTree SingleLeft(AVLTree A) {    /*左单旋*/     AVLTree B;      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;  }    AVLTree SingleRight(AVLTree A) {    /*右单旋*/      AVLTree B;      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->Right), A->Height) + 1;       return B;  }    AVLTree DoubleLeftRight(AVLTree A) {     /*右-左单旋*/     A->Left = SingleRight(A->Left);       return SingleLeft(A);  }    AVLTree DoubleRightLeft(AVLTree A) {    /*左-右单旋*/     A->Right = SingleLeft(A->Right);      return SingleRight(A);  }