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04-树5 Root of AVL Tree

浙大mooc 站点默认 1个月前 (11-09) 13次浏览 已收录 0个评论

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.
8006cf557f9b2b0c01580ac9a31cd849.png

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

题意没什么好说的,一看就明白了。这题最主要是让我们学会平衡二叉树在插入节点的时候,如果平衡因子大于1了,这这时我们如何去写出代码将树翻转保持原来的平衡因子。而难点也在于这里,但是mooc课件中就已经给出了平衡翻转的代码,虽然只给出了ll和lr的但是我们模仿也很容易写出rr和rl 的,但是我们不能仅仅模仿写完这题就完事了,更需要从深层理解的角度去明白是如何翻转的,能够自己写出这四个翻转函数

#include<iostream>
#include<malloc.h>
using namespace std;
typedef int ElementType;
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;
}
// 返回树高,空树返回 -1 
int GetHeight(AVLTree A) {
	return A == NULL ? -1 : A->Height;
}
AVLTree SingleRightRotation(AVLTree A);
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 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 SingleRightRotation(AVLTree A) {
	// 此时根节点是 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->Left), A->Height) + 1;
	return B;  // 此时 B 为根结点了 
}
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,tmp;
	cin >> n;
	AVLTree a=NULL;
	for (int i = 0; i < n; i++) {
		cin >> tmp;
		a = Insert(a, tmp);
	}
	cout << a->Data;
}

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