C++ 数据结构经典题目AVL树的创建 Root of AVL Tree （25 point(s))

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

Author: 陈越

Organization: 浙江大学

Time Limit: 400 ms

Memory Limit: 64 MB

Code Size Limit: 16 KB

平衡二叉树AVL树的定义是左边的高度和右边的高度不超过1(<=1),为了维持二叉搜索树的平衡性，常见的操作有四种，左单旋，右单旋，左右旋，右左旋

如果插在二叉树的左子数，即小于二叉树的值

需要左旋的情况是，GetHeight(T->Left) - GetHeight(T->Right) == 2          // GeiHegiht返回当前节点的高度

如果此时X < T->Left->Data，则是左单旋

否的话，就是左右双旋

如果插在二叉树的右子数，即大于二叉树的值

且满足GetHeight(T->Left) - GetHeight(T->Right) == -2 需要右旋

如果X > T->Right->Data ,右单旋即可，若否，则需要右左双旋

#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 */
};

int Max(int a, int b)
{
	return a > b ? a : b;
}

int GetHeight(AVLTree A) {
	if (!A)
		return -1;
	return A->Height;
}

AVLTree SingleLeftRotation(AVLTree A)
{ /* 注意：A必须有一个左子结点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)
{
	AVLTree B = A->Right;
	A->Right = B->Left;
	B->Left = A;
	A->Height = Max(GetHeight(A->Left), GetHeight(A->Right)) + 1;   // 跟新A的高度
	B->Height = Max(GetHeight(B->Left), A->Height) + 1;             // 跟新B的高度
	return B;
}

AVLTree DoubleLeftRightRotation(AVLTree A)
{ /* 注意：A必须有一个左子结点B，且B必须有一个右子结点C */
  /* 将A、B与C做两次单旋，返回新的根结点C */

	/* 将B与C做右单旋，C被返回 */
	A->Left = SingleRightRotation(A->Left);
	/* 将A与C做左单旋，C被返回 */
	return SingleLeftRotation(A);
}

/*************************************/
/* 对称的右单旋与右-左双旋请自己实现 */
AVLTree DoubleRightLeftRotation(AVLTree A)
{
	A->Right = SingleLeftRotation(A->Right);
	return SingleRightRotation(A);
}



/*************************************/

AVLTree Insert(AVLTree T, ElementType X)
{ /* 将X插入AVL树T中，并且返回调整后的AVL树 */
	if (!T) { /* 若插入空树，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;
	} /* if (插入空树) 结束 */

	else if (X < T->Data) {
		/* 插入T的左子树 */
		T->Left = Insert(T->Left, X);
		/* 如果需要左旋 */
		if (GetHeight(T->Left) - GetHeight(T->Right) == 2)
			if (X < T->Left->Data)
				T = SingleLeftRotation(T);      /* 左单旋 */
			else
				T = DoubleLeftRightRotation(T); /* 左-右双旋 */
	} /* else if (插入左子树) 结束 */

	else if (X > T->Data) {
		/* 插入T的右子树 */
		T->Right = Insert(T->Right, X);
		/* 如果需要右旋 */
		if (GetHeight(T->Left) - GetHeight(T->Right) == -2)
			if (X > T->Right->Data)
				T = SingleRightRotation(T);     /* 右单旋 */
			else
				T = DoubleRightLeftRotation(T); /* 右-左双旋 */
	} /* else if (插入右子树) 结束 */

	/* else X == T->Data，无须插入 */

	/* 别忘了更新树高 */
	T->Height = Max(GetHeight(T->Left), GetHeight(T->Right)) + 1;

	return T;
}

int main()
{
	int n, temp;
	AVLTree root = NULL;
	scanf("%d", &n);
	for (int i = 0; i < n; i++) {
		scanf("%d", &temp);
		root = Insert(root, temp);
	}
	printf("%d\n", root->Data);
}