數據結構-平衡二叉樹（AVL Tree）

在 數據結構-二叉樹(binary tree)-二叉查找樹(binary search tree) 的最後面，提到過在二叉樹中增加或者刪除節點，可能導致樹的左右子樹高度相差很多，即導致樹不平衡。爲了解決這個問題，規定在插入或者刪除節點的時候，必須保證每一個節點的左右子樹的高度差的絕對值不超過1，| height(left) - height(right) | <= 1。這樣的二叉樹稱爲平衡二叉樹（AVL Tree）。定義左右子樹的高度差的絕對值爲平衡因子。

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

判斷AVL Tree

根絕平衡二叉樹的定義，可以很容易實現判斷二叉樹是否爲平衡二叉樹

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));
}

這個方法通過先計算左右子樹的高度，再判斷二叉樹是否爲二叉樹。方便理解，但是這種方法訪問了很多次某些節點，特別是越接近葉節點的節點。在計算整棵樹的高度時訪問了所有的節點，接着又分別計算以根節點的左右子節點爲根節點的二叉樹高度，再次訪問了除根節點以外的所有節點，如此導致大量重複訪問。一棵平衡二叉樹的任意一個子樹也都是一個平衡二叉樹，因此可以考慮從下往上遍歷，如果發現一棵子樹不是平衡二叉樹，那麼整棵樹肯定不是平衡二叉樹，如此減少大量的重複訪問。

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;
}

當插入或者刪除操作破壞了二叉樹的平衡性，需要進行一些相關操作，將其調整爲平衡二叉樹，相關操作包括：單旋轉和雙旋轉。

單旋轉-右旋（RR）

在B的左子樹中插入新元素後，導致A節點的平衡因子變爲2，二叉樹不再平衡。以B節點爲中心向右旋轉，使得B節點上升一層，A節點下降一層，從而將二叉樹重新調整爲平衡二叉樹。

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;
}

單旋轉-左旋（LL）

左旋操作與右旋操作相似：

//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;
}

雙旋-左旋+右旋(LR)

考慮如下圖的情況，此時右單旋轉不能解決問題，問題從A節點不平衡轉移到了B節點不平衡。

所以要進行左右雙旋轉(LR)。

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

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

    return SingleRotateright (node1);

}

雙旋-右旋+左旋（RL）

與上述情況類似，在某些情況下需要進行RL操作。

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

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

    return SingleRotateLeft (node1);

}

綜合所有操作

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);

}