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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 (≤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
具体代码实现为:
/*********************平衡二叉树***************************/
#include<stdio.h>
#include<stdlib.h>
#define ElementType int
typedef struct AVLNode *Position;
typedef Position AVLTree; /* AVL树类型 */
struct AVLNode{
ElementType Data; /* 结点数据 */
AVLTree Left; /* 指向左子树 */
AVLTree Right; /* 指向右子树 */
int Height; /* 树高 */
};
AVLTree AVL_Insertion(ElementType X,AVLTree T);
int main()
{
int x,N;
AVLTree T = NULL;
scanf("%d",&N);
for(int i=0; i<N; i++){
scanf("%d",&x);
T = AVL_Insertion(x,T);
}
printf("%d\n",T->Data);
return 0;
}
int GetHeight(AVLTree BT)
{
int HL,HR,MaxH;
if( BT ){
HL = GetHeight(BT->Left); //获得左子树的高度
HR = GetHeight(BT->Right); //获得右子树的高度
MaxH = HL > HR ? HL : HR; //取左右子树中较大的高度
return (MaxH+1); //返回树的高度
}else{
return 0;
}
}
int Max( int a, int b )
{
return (a > b ? a : b);
}
AVLTree SingleLeftRotation(AVLTree A)
{
/*注意:A必须有一个左儿子结点B*/
/* 将A与B做左单旋,更新A与B的高度,返回新的根结点B */
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;
}
AVLTree SingleRightRotation(AVLTree A)
{
/*注意:A必须有一个右儿子结点B*/
/* 将A与B做右单旋,更新A与B的高度,返回新的根结点B */
AVLTree B = A->Right;
A->Right = B->Left;
B->Left = A;
A->Height = Max( GetHeight(A->Left),GetHeight(A->Right) )+1;
B->Height = Max( A->Height,GetHeight(B->Right) )+1;
return B;
}
AVLTree DoubleLeftRightRotation(AVLTree A)
{
/*注意:A必须有一个左儿子结点B,且B必须有一个右儿子结点C*/
/* 将A、B与C做两次单旋,返回新的根结点C */
A->Left = SingleRightRotation(A->Left); //将B与C做右单旋,C被返回
return SingleLeftRotation(A); //将A与C做左单旋,C被返回
}
AVLTree DoubleRightLeftRotation(AVLTree A)
{
/*注意:A必须有一个右儿子结点B,且B必须有一个左儿子结点C*/
/* 将A、B与C做两次单旋,返回新的根结点C */
A->Right = SingleLeftRotation(A->Right); //将B与C做左单旋,C被返回
return SingleRightRotation(A); //将A与C做右单旋,C被返回
}
AVLTree AVL_Insertion(ElementType X,AVLTree T)
{
/*将X插入AVL树T中,并且返回调整后的AVL树*/
if( !T ){/*若插入的是空树,则新建包含一个结点的树*/
T = (AVLTree)malloc(sizeof(struct AVLNode));
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){/*需要左旋*/
if(X < T->Left->Data){
T = SingleLeftRotation(T); //左单旋
}else{
T = DoubleLeftRightRotation(T); //左-右双旋
}
}
}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 ){/*需要右旋*/
if(X > T->Right->Data){
T = SingleRightRotation(T); //右单旋
}else{
T = DoubleRightLeftRotation(T); //右-左双旋
}
}
}
/*else X==T->Data,无须插入*/
T->Height = Max( GetHeight(T->Left),GetHeight(T->Right) ); //更新树高
return T;
}