题目描述

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.

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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

  • 构造平衡二叉树即可

代码

很多地方直接copy了浙大数据结构mooc的代码-_-

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#include<iostream>
#define ElementType int
using namespace std;

typedef struct AVLNode * Position;
typedef Position AVLTree;/* AVL树类型 */
struct AVLNode {
	ElementType Data;
	AVLTree Left;
	AVLTree Right;
	int Height;
};

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

int GetHeight(AVLTree A) {
	if(A)
		return Max(GetHeight(A->Left), GetHeight(A->Right)) + 1;
	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) {
	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->Right), A->Height) + 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) {
	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;
	ElementType X;
	AVLTree root=NULL;

	cin >> N;
	for (int i = 0; i < N; i++) {
		cin >> X;
		root = Insert(root, X);
	}
	cout << root->Data;
	return 0;
}