浙大数据结构的编程题——Root of AVL Tree

数据结构-Root of AVL Tree

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:

Sample Input 2:

7
88 70 61 96 120 90 65

Sample Output 2:

思路

本题主要分为两部分内容，一是插入结点，二是检查结点插入后树是否平衡，若不平衡，则通过LL、RR、LR、RL旋转使树重新平衡。

插入结点

此处可以参考是否同一颗二叉搜索树这题的思路与code.

检查结点

检查结点在插入之后，插入完成时，程序递归返回，顺势检查返回路径是否有失衡问题，即左右子树高度相差绝对值为2的情况。然后判断失衡情况的类型：

插入结点在发现失衡的结点的左子树的左侧 -> LL旋转
插入结点在发现失衡的结点的左子树的右侧 -> LR旋转
插入结点在发现失衡的结点的右子树的左侧 -> RL旋转
插入结点在发现失衡的结点的右子树的右侧 -> RR旋转

L也叫左旋，为顺时针旋转。

R同理。

以LR旋转为例，插入结点在发现失衡的结点的左子树的右侧，则发现结点的左子树要先进行一次右旋，再在发现结点进行一次左旋。

RL同理。

Code

#include <iostream>
#include <string>
#include <vector>
#include <list>
#include <map>
#include <algorithm>
#include <iterator>
#include <math.h>
#include <stack>
#include <queue>
using namespace std;

#define ElementType int
typedef struct AVLNode *Position;
typedef Position AVLTree; /* AVL树类型 */
AVLTree SingleLeftRotation(AVLTree A);

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 T) {
	if(T)
		return T->Height;
	else 
		return -1;
}

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)
{
	//右单旋
	AVLTree 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->Left), GetHeight(B->Right)) + 1;
	
	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);
}

AVLTree DoubleRightLeftRotation(AVLTree A)
{ 
	//插入结点在发现不平衡的结点的右子树的左侧

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


/*************************************/
/* 对称的右单旋与右-左双旋请自己实现 */
/*************************************/

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);
		//先插入，再检查 
		/* 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() {
	//begin:			end:
	ios::sync_with_stdio(false);
	int N;
	ElementType data;
	AVLTree T=NULL; 

	scanf("%d\n", &N);
	for (int i = 0; i < N; i++) {
		scanf("%d", &data);
		T = Insert(T, data);
	}
	if (T)
		printf("%d", T->Data);


	return 0;
}

数据结构 Root_of_AVL_Tree