平衡树（AVL）实现时注意的一些东西-优快云博客

本文介绍了AVL树的基本概念及插入操作时的平衡调整方法，包括四种旋转操作：单次左旋、先右旋再左旋、单次右旋和先左旋再右旋。并提供了AVL树的C语言实现代码示例。

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1.概念

1.AVL树必须满足是一棵二叉查找（排序）树

2.任意节点左右子树高度差不能超过1

2.插入操作

插入操作有可能让树不满足第二条规则，那么必须旋转树。有四种旋转方式

1.单次左旋（左旋可以理解为旋转左边的孩子，右旋同理） ----> 在节点的左孩子左子树上插入，高度差 = 2

2.先右旋，再左旋 ----> 在节点的左孩子右子树上插入，高度差 = 2

3.单次右旋 ----> 在节点的右孩子右子树上插入，高度差 = -2

4.先左旋，再右旋 ----> 在节点的右孩子左子树上插入，高度差 = -2

记住，插入操作的结尾处，必须要更新当前节点的高度，方法为max(left->height,right->height)+1

以下附上自己实现的一个AVL树源码（remove操作还没有完成，不好意思）

avl_tree.h

#ifndef __AVL_TREE_H__
#define __AVL_TREE_H__

#include <malloc.h>
#include <stdlib.h>
#include <assert.h>
#include <math.h>

typedef struct _unode
{
	int id;
//	char* name;
//	char* pass;
}unode;

int isLess(const unode* lhs,const unode* rhs)
{
	return lhs->id < rhs->id;
}

int isGreat(const unode* lhs,const unode* rhs)
{
	return lhs->id > rhs->id;
}

int isEqual(const unode* lhs,const unode* rhs)
{
	return lhs->id == rhs->id;
}

typedef struct _tnode
{
	unode* data;
	struct _tnode* left;
	struct _tnode* right;
	int height;
}tnode;

void singleRotateLeft(tnode** root);
void doubleRotateLeft(tnode** root);
void singleRotateRight(tnode** root);
void doubleRotateRight(tnode** root);

void singleRotateLeft(tnode** root)
{
	tnode* k1 = (*root)->left;
	(*root)->left = k1->right;
	k1->right = *root;
	(*root)->height = max(height((*root)->left),height((*root)->right)) + 1;
	k1->height = max(height(k1->left),height(k1->right)) + 1;
	*root = k1;
}

void doubleRotateLeft(tnode** root)
{
	singleRotateRight(&(*root)->left);
	singleRotateLeft(root);
}

void singleRotateRight(tnode** root)
{
	tnode* k1 = (*root)->right;
	(*root)->right = k1->left;
	k1->left = *root;
	(*root)->height = max(height((*root)->left),height((*root)->right)) + 1;
	k1->height = max(height(k1->left),height(k1->right)) + 1;
	*root = k1;
}

void doubleRotateRight(tnode** root)
{
	singleRotateLeft(&(*root)->right);
	singleRotateRight(root);
}


tnode* create_avl_tree(const unode* p)
{
	//tnode* root = (tnode*)malloc(sizeof(tnode));
	//root->data = p;
	//root->left = root->right = NULL;
	//root->height = 0;
	//return root;
	return NULL;
}

int height(const tnode* node)
{
	if(node == NULL)
	{
		return -1;
	}
	return node->height;
}

void insert(tnode** root,tnode* data)
{
	tnode* r = *root;
	if( *root == NULL )
	{
		*root = data;
	}
	else if( isGreat( (*root)->data, data->data ) )
	{
		insert( &((*root)->left), data );
		if(height((*root)->left) - height((*root)->right) == 2) // ready to rotate
		{
			if(isLess(data->data,(*root)->left->data))
			{
				printf("singleRotateLeft ==> %d\n",data->data->id);
				singleRotateLeft(&(*root));
			}
			else
			{
				printf("doubleRotateLeft ==> %d\n",data->data->id);
				doubleRotateLeft(&(*root));
			}
		}
	}
	else if( isLess( (*root)->data, data->data ) )
	{
		insert( &((*root)->right), data );
		if(height((*root)->left) - height((*root)->right) == -2) // ready to rotate
		{
			if(isGreat( data->data,(*root)->right->data))
			{
				printf("singleRotateRight ==> %d\n",data->data->id);
				singleRotateRight(&(*root));
			}
			else
			{
				printf("doubleRotateRight ==> %d\n",data->data->id);
				doubleRotateRight(&(*root));
			}
		}
	}
	else if( isEqual( (*root)->data, data->data ) )
	{
		assert(0);
	}
	(*root)->height = max(height((*root)->left),height((*root)->right)) + 1;
}

tnode* find(tnode* root,const unode* id)
{
	if(root != NULL)
	{
		if(isEqual(root->data, id))
		{
			return root;
		}
		else if(isLess(id, root->data))
		{
			return find(root->left,id);
		}
		else if(isGreat(id, root->data))
		{
			return find(root->right,id);
		}
	}
	return NULL;
}

void remove(tnode** root,const unode* id)
{
	tnode* p = NULL;
	tnode* q = *root;
	while(q != NULL)
	{
		if(isEqual(q->data, id))
		{
			break;
		}
		else if(isLess(id, q->data))
		{
			p = q;
			q = q->left;
		}
		else if(isGreat(id, q->data))
		{
			p = q;
			q = q->right;
		}
	}

	if(q->left == NULL && q->right == NULL) // 无子节点
	{
		if(p != NULL) // 删除的节点非根节点
		{
			if(isLess(q->data,p->data))
			{
				p->left = NULL;
			}
			else
			{
				p->right = NULL;
			}
			free(q);
		}
		else // 删除的节点为根节点
		{
			*root = NULL;
			free(q);
		}
	}
	else if(q->left == NULL && q->right != NULL) // 只有右儿子
	{
		if(p != NULL) // 删除的节点非根节点
		{
			if(isLess(q->data,p->data))
			{
				p->left = q->right;
			}
			else
			{
				p->right = q->right;
			}
			free(q);
		}
		else
		{
			*root = q->right;
			free(q);
		}
	}
	else if(q->left != NULL && q->right == NULL) // 只有左儿子
	{
		if(p != NULL) // 删除的节点非根节点
		{
			if(isLess(q->data,p->data))
			{
				p->left = q->left;
			}
			else
			{
				p->right = q->left;
			}
			free(q);
		}
		else
		{
			*root = q->left;
			free(q);
		}
	}
	else if(q->left != NULL && q->right != NULL)
	{
		// 查找右子树中最小的节点，并进行替换
		tnode* minNode = q->right;
		unode* tmp = (unode*)malloc(sizeof(unode));
		while(minNode->left != NULL)
		{
			minNode = minNode->left;
		}
		memcpy(tmp,minNode->data,sizeof(unode)); 
		// 删除那个最小的节点
		remove(&q,minNode->data);
		q->data = tmp;
	}
}


void printTreeBefore(const tnode* root,int layer)
{
	int t = layer;
	if(root != NULL)
	{
		while(t-- > 0)
		{
			printf("  ");
		}
		printf("--> %d  at %d\n",root->data->id,root->height);
		printTreeBefore(root->left,layer + 1);
		printTreeBefore(root->right,layer + 1);
	}
}

void printTreeMiddle(const tnode* root,int layer)
{
	int t = layer;
	if(root != NULL)
	{
		printTreeMiddle(root->left,layer + 1);
		while(t-- > 0)
		{
			printf("  ");
		}
		printf("--> %d  at %d\n",root->data->id,root->height);
		printTreeMiddle(root->right,layer + 1);
	}
}

void printTreeAfter(const tnode* root,int layer)
{
	int t = layer;
	if(root != NULL)
	{
		printTreeAfter(root->left,layer + 1);
		printTreeAfter(root->right,layer + 1);
		while(t-- > 0)
		{
			printf("  ");
		}
		printf("--> %d  at %d\n",root->data->id,root->height);
	}
}


#endif

测试文件

#include "avltree.h"


int main()
{
	int n = 0;
	int i = 0;
	//int dlist[] = {144,20,235,83,64,253,246,277,25,296};
	//int dlist[] = {10,12,14,16};
	//int dlist[] = {10,8,6,4};
	int dlist[] = {50,20,30,25,35,32};
	tnode* root;
	tnode* p1 = NULL;
	unode* d = (unode*)malloc(sizeof(unode));
	d->id = 30;
	root = create_avl_tree(d);
	srand(time(0));


	for(;i < _countof(dlist);i++)
	{
		tnode* p = (tnode*)malloc(sizeof(tnode));
		p->data = (unode*)malloc(sizeof(unode));
		//p->data->id = (int)(rand() / 100.0);
		p->data->id = dlist[i];
		p->left = p->right = NULL;
		insert(&root,p);
		printf("--> %d\n",p->data->id);
	}


// 	p1 = find(root,54);
// 	p1 = find(root,246);
// 	p1 = find(root,83);
// 	p1 = find(root,235);

	printf("-------------------\n");
	printTreeBefore(root,0);

	remove(&root,d);

	printf("-------------------\n");
	printTreeBefore(root,0);

	printf("-------------------\n");
	printTreeMiddle(root,0);

	printf("-------------------\n");
	printTreeAfter(root,0);
	n = 11100;
}