定义：
- 左子树与右子树的高度之差的绝对值小于等于1
- 左右子树也是平衡二叉树
a.平衡因子定义: 节点的左子树与右子树的深度之差**
例如：
平衡树的平衡方法:
（1）LL型
B = A->lchild;
A->lchild = B->rchild;
B->rchild = A;
A->bf = 0;
B->bf = 0;
if ( FA == NULL )
{
*root = B;
}
else if (A == FA->lchild )
{
FA->lchild = B;
}
else
{
FA->rchild = B;
}（2）LR型
B = A->lchild;
PAVLNode C = B->rchild;
B->lchild = C->lchild;
A->lchild = C->rchild;
C->lchild = B;
C->rchild = A;
if (key > C->key)
{
A->bf = 0;
B->bf = 1;
C->bf = 0;
}
if (key < C->key)
{
A->bf = -1;
B->bf = 0;
C->bf = 0;
}
if (FA == NULL)
{
*root = C;
}
else if (FA->lchild == A)
{
FA->lchild = C;
}
else
{
FA->rchild = C;
}（3）RR 型
B = A->rchild;
A->rchild = B->lchild;
B->lchild = A;
A->bf = 0;
B->bf = 0;
if (FA == NULL)
{
*root = B;
}
else if (FA->lchild == A)
{
FA->lchild = B;
}
else
{
FA->rchild = B;
}（4）RL 型
B = A->rchild;
PAVLNode C = B->lchild;
A->rchild = C->lchild;
B->lchild = C->rchild;
C->lchild = A;
C->rchild = B;
if (C->key < key)
{
A->bf = 1;
B->bf = 0;
C->bf = 0;
}
else
{
A->bf = 0;
B->bf = -1;
C->bf = 0;
}
if (FA == NULL)
{
*root = C;
}
else if (FA->lchild == A)
{
FA->lchild = C;
}
else
{
FA->rchild = C;
}平衡树建立方法：
- 查找应插入位置，同时记录距离插入位置最近的可能失衡点A(A的平衡因子不等于0)
- 插入新节点S
- 确定节点B,并修改A的平衡因子，
- 修改从B到S路径上的个节点的平衡因子（原值必为0）
- 根据A，B的平衡因子判断失衡以及失衡类型，并做相应处理
代码实现：
typedef struct tree
{
int bf;//平衡因子
int key;//键值
struct tree * lchild;//左孩子
struct tree * rchild;//右孩子
}AVLNode,*PAVLNode;
void inser_AVLTree(PAVLNode * root, int key)
{
PAVLNode fa = NULL, p = *root, A = *root,B = NULL, fp = NULL,FA = NULL;
PAVLNode pNew = new AVLNode;
pNew->bf = 0;
pNew->key = key;
pNew->lchild = pNew->rchild = NULL;
//判断是否为空
if (NULL == *root)
{
(*root) = pNew;
}
/*如果不为空则进行插入*/
else
{
/*查找插入点fp*/
while (p)
{
/*找到距离插入位置最近的可能失衡点A（A的平衡因子不等于0）*/
if ( p->bf != 0 )
{
A = p;
FA = fp;
}
fp = p;
if (p->key < key)
{
p = p->rchild;
}
else
{
p = p->lchild;
}
}
/*插入pNew*/
if (fp->key > key)
{
fp->lchild = pNew;
}
else
{
fp->rchild = pNew;
}
/*调节平衡因子*/
if (key > A->key)
{
B = A->rchild;
A->bf -= 1;
}
if (key < A->key)
{
B = A->lchild;
A->bf += 1;
}
/*修改pNew 到 A路径上的平衡因子*/
p = B;
while (p != pNew)
{
if (p->key < key)
{
p->bf = -1;
p = p->rchild;
}
else
{
p->bf = 1;
p = p->lchild;
}
}
/*判断失衡并做相应处理*/
//LL型
if (A->bf == 2 && B->bf == 1)
{
B = A->lchild;
A->lchild = B->rchild;
B->rchild = A;
A->bf = 0;
B->bf = 0;
if ( FA == NULL )
{
*root = B;
}
else if (A == FA->lchild )
{
FA->lchild = B;
}
else
{
FA->rchild = B;
}
}
//LR型
else if (A->bf == 2 && B->bf == -1)
{
B = A->lchild;
PAVLNode C = B->rchild;
B->lchild = C->lchild;
A->lchild = C->rchild;
C->lchild = B;
C->rchild = A;
if (key > C->key)
{
A->bf = 0;
B->bf = 1;
C->bf = 0;
}
if (key < C->key)
{
A->bf = -1;
B->bf = 0;
C->bf = 0;
}
if (FA == NULL)
{
*root = C;
}
else if (FA->lchild == A)
{
FA->lchild = C;
}
else
{
FA->rchild = C;
}
}
//RL型
else if (A->bf == -2 && B->bf == 1)
{
B = A->rchild;
PAVLNode C = B->lchild;
A->rchild = C->lchild;
B->lchild = C->rchild;
C->lchild = A;
C->rchild = B;
if (C->key < key)
{
A->bf = 1;
B->bf = 0;
C->bf = 0;
}
else
{
A->bf = 0;
B->bf = -1;
C->bf = 0;
}
if (FA == NULL)
{
*root = C;
}
else if (FA->lchild == A)
{
FA->lchild = C;
}
else
{
FA->rchild = C;
}
}
//RR型
else if (A->bf == -2 && B->bf == -1)
{
B = A->rchild;
A->rchild = B->lchild;
B->lchild = A;
A->bf = 0;
B->bf = 0;
if (FA == NULL)
{
*root = B;
}
else if (FA->lchild == A)
{
FA->lchild = B;
}
else
{
FA->rchild = B;
}
}
}
}举例
已知如下所示长度为12的表 (5,4,8,1,9,7,6,2,12,11,10,3)按所给顺序构造二叉平衡树
