a. AVL树有一个性质，F(h)=F(h-1)+F(h-2)+1

设斐波那契数列=h(x)，则F(h)=h(h+2)-1，参考3.2节斐波那契部分可知，斐波那契数以指数形式增长，h(x)=h^x/5^1/2

所以h=O(lgn)

b.c.d.

#include <stdio.h>
#include <stdlib.h>

typedef struct AVLNode
{
	int key;
	int hight;
	AVLNode *p;
	AVLNode *left;
	AVLNode *right;
}AVL,*pAVL;

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

int getHeight(pAVL x)
{
	if(x==NULL)
		return 0;
	else 
		return x->hight;
}

void updateHight(pAVL x)
{
	if(x)
	{
		x->hight=max(getHeight(x->left),getHeight(x->right))+1;
	}
}

pAVL leftRotate(pAVL x)
{
	if(x)
	{
		pAVL y=x->right;
		if(!y)
			return NULL;
		if(x->p)
		{
			if(x==x->p->left)
				x->p->left=y;
			else
				x->p->right=y;
		}
		y->p=x->p;		
		x->right=y->left;
		y->left=x;
		x->p=y;
		updateHight(x);
		updateHight(y);
		return y;
	}
	return NULL;
}

pAVL rightRotate(pAVL y)
{
	if(y)
	{
		pAVL x=y->left;
		if(!x)
			return NULL;
		if(y->p)
		{
			if(y==y->p->left)
				y->p->left=x;
			else
				y->p->right=x;
		}
		x->p=y->p;
		y->left=x->right;
		x->right=y;
		y->p=x;
		updateHight(y);
		updateHight(x);
		return x;
	}
	return NULL;
}

pAVL newNode(int key)
{
	pAVL x=(pAVL)malloc(sizeof(AVL));
	x->hight=1;
	x->key=key;
	x->left=NULL;
	x->right=NULL;
	x->p=NULL;
	return x;
}

pAVL balance(pAVL x,int newKey)
{
	if(getHeight(x->left)-getHeight(x->right)>=2)
	{
		if(newKey<=x->left->key)
		{
			return rightRotate(x);
		}
		else
		{
			leftRotate(x->left);
			return rightRotate(x);
		}
	}
	if(getHeight(x->right)-getHeight(x->left)>=2)
	{
		if(newKey>x->right->key)
		{
			return leftRotate(x);
		}
		else
		{
			rightRotate(x->right);
			return leftRotate(x);
		}
	}
	return x;
}

void AVLinsert(pAVL *root,int key)
{
	pAVL z=newNode(key),x=*root,y=NULL;	
	//普通二叉搜索树插入
	while(x)
	{
		y=x;
		if(key<=x->key)
			x=x->left;
		else
			x=x->right;
	}
	if(y==NULL)
	{
		(*root)=z;
		return;
	}
	z->p=y;
	if(key<=y->key)
	{
		y->left=z;
	}
	else
	{
		y->right=z;
	}
	//从插入节点开始向上更新高度
	pAVL i=z->p;
	while(i)
	{
		updateHight(i);
		i=i->p;
	}
	//从插入节点的祖父开始向上检查平衡情况
	i=z;
	if(!i->p || !i->p->p)
		return;
	i=i->p->p;
	while(i)
	{
		i=balance(i,key);
		if(i->p==NULL)
			(*root)=i;
		i=i->p;
	}
}

void preTrav(pAVL root,char c)
{
	pAVL x=root;
	if(!x)
		return;
	printf("key=%d,di=%c,height=%d ",x->key,c,x->hight);
	preTrav(x->left,'L');
	preTrav(x->right,'R');
}

void main()
{
	pAVL root=NULL;
	AVLinsert(&root,50);
	AVLinsert(&root,40);
	AVLinsert(&root,60);
	AVLinsert(&root,55);
	AVLinsert(&root,70);
	AVLinsert(&root,53);
	preTrav(root,'M');
	getchar();
}