在 数据结构-二叉树(binary tree)-二叉查找树(binary search tree) 的最后面，提到过在二叉树中增加或者删除节点，可能导致树的左右子树高度相差很多，即导致树不平衡。为了解决这个问题，规定在插入或者删除节点的时候，必须保证每一个节点的左右子树的高度差的绝对值不超过1，| height(left) – height(right) | <= 1。这样的二叉树称为平衡二叉树（AVL Tree）。定义左右子树的高度差的绝对值为平衡因子。

来源：http://www.cnblogs.com/zhuwbox/p/3636783.html

判断AVL Tree

根绝平衡二叉树的定义，可以很容易实现判断二叉树是否为平衡二叉树

bool IsBalanced1 (Node *root)
{
    if (root == NULL) return true;

    int left = TreeHeight(root->left);
    int right = TreeHeight(root->right);
    int diff = left - right;
    if (diff < -1 || diff > 1)
        return false;
    else
        return (IsBalanced1(root->left) && IsBalanced1(root->left));
}

这个方法通过先计算左右子树的高度，再判断二叉树是否为二叉树。方便理解，但是这种方法访问了很多次某些节点，特别是越接近叶节点的节点。在计算整棵树的高度时访问了所有的节点，接着又分别计算以根节点的左右子节点为根节点的二叉树高度，再次访问了除根节点以外的所有节点，如此导致大量重复访问。一棵平衡二叉树的任意一个子树也都是一个平衡二叉树，因此可以考虑从下往上遍历，如果发现一棵子树不是平衡二叉树，那么整棵树肯定不是平衡二叉树，如此减少大量的重复访问。

bool IsBalanced2 (Node *root, int& depth)
{
    if (root == NULL) {
        depth = 0;
        return true;
    }

    int left, right;
    if (IsBalanced2(root->left, left) && IsBalanced2(root->right, right)) {
        int diff = left - right;
        if (diff >= -1 && diff <= 1) {
            depth = 1 + (left > right ? left : right);
            return true;
        }
    }
    return false;
}

当插入或者删除操作破坏了二叉树的平衡性，需要进行一些相关操作，将其调整为平衡二叉树，相关操作包括：单旋转和双旋转。

单旋转-右旋（RR）

在B的左子树中插入新元素后，导致A节点的平衡因子变为2，二叉树不再平衡。以B节点为中心向右旋转，使得B节点上升一层，A节点下降一层，从而将二叉树重新调整为平衡二叉树。

Node* SingleRotateRight (Node* node1) {

    Node *node2 = node1->right;
    node1->right = node2->left;
    node2->left = node1;

    node1->height = max ( Height (node1->left), Height (node1->right)) + 1;
    node2->height = max ( Height (node2->right), node1->height) + 1;

    return node2;
}

单旋转-左旋（LL）

左旋操作与右旋操作相似：

//node1: node which is not balanced
Node* SingleRotateLeft (Node* node1) {

    Node *node2 = node1->left;
    node1->left = node2->right;
    node2->right = node1;

    node1->height = max ( Height (node1->left), Height (node1->right)) + 1;
    node2->height = max ( Height (node2->left), node1->height) + 1;

    return node2;
}

双旋-左旋+右旋(LR)

考虑如下图的情况，此时右单旋转不能解决问题，问题从A节点不平衡转移到了B节点不平衡。

所以要进行左右双旋转(LR)。

// LR rotate
Node* DoubleRotateRight (Node *node1) {

    node1->right = SingleRotateLeft (node1->right);

    return SingleRotateright (node1);

}

双旋-右旋+左旋（RL）

与上述情况类似，在某些情况下需要进行RL操作。

//RL rotate
Node* DoubleRotateLeft (Node *node1) {

    node1->left = SingleRotateRight (node1->left);

    return SingleRotateLeft (node1);

}

综合所有操作

typedef struct node
{
    int val;
    int height;
    struct node *left;
    struct node *right;
} Node;

bool IsBalanced1 (Node *root)
{
    if (root == NULL) return true;

    int left = TreeHeight(root->left);
    int right = TreeHeight(root->right);
    int diff = left - right;
    if (diff < -1 || diff > 1)
        return false;
    else
        return (IsBalanced1(root->left) && IsBalanced1(root->left));
}

bool IsBalanced2 (Node *root, int& depth)
{
    if (root == NULL) {
        depth = 0;
        return true;
    }

    int left, right;
    if (IsBalanced2(root->left, left) && IsBalanced2(root->right, right)) {
        int diff = left - right;
        if (diff >= -1 && diff <= 1) {
            depth = 1 + (left > right ? left : right);
            return true;
        }
    }
    return false;
}


void Insert (Node* root, int val) {

    if (root == NULL) {

        root = (Node*)malloc(sizeof(Node));
        if (root == NULL) {
            printf("space error!\n");
            return root;
        } else {
            root->val = val;
            root->left = root->right = NULL;
            root->height = 0;
        }
    } else if (val < root->val){
        root->left  = Insert (root->left, val);
        if (Height (root->left) - Height (root->right) == 2) {
            if (val < root->left->val)
                root = SingleRotateLeft (root);
            else
                root = DoubleRotateLeft (root);
        }

    } else if (val > root->val) {
        root->right = Insert (root->right, val);
        if (Height(root->right) - Height (root->left) == 2) {
            if (val > root->right->val)
                root = SingleRotateRight (root);
            else
                root = DoubleRotateRight (root);
        }
    }
}

int Height (Node* root) {
    if(root == NULL)
        return -1;
    else
        return root->height;
}

//node1: node which is not balanced
Node* SingleRotateLeft (Node* node1) {

    Node *node2 = node1->left;
    node1->left = node2->right;
    node2->right = node1;

    node1->height = max ( Height (node1->left), Height (node1->right)) + 1;
    node2->height = max ( Height (node2->left), node1->height) + 1;

    return node2;
}

//RL rotate
Node* DoubleRotateLeft (Node *node1) {

    node1->left = SingleRotateRight (node1->left);

    return SingleRotateLeft (node1);

}

Node* SingleRotateRight (Node* node1) {

    Node *node2 = node1->right;
    node1->right = node2->left;
    node2->left = node1;

    node1->height = max ( Height (node1->left), Height (node1->right)) + 1;
    node2->height = max ( Height (node2->right), node1->height) + 1;

    return node2;
}

// LR rotate
Node* DoubleRotateRight (Node *node1) {

    node1->right = SingleRotateLeft (node1->right);

    return SingleRotateright (node1);

}