一、平衡二叉树的定义

1. 使树的高度在每次插入元素后仍然能保持O(logn)的级别
2. AVL仍然是一棵二叉查找树
3. 左右子树的高度之差是平衡因子,且值不超过1

//数据类型
struct node{
    int v, height;
    node *lchild, *rchild;
};

//新建一个结点
node* newNode(int v){
    node* Node = new node;
    Node->v = v;
    Node->height = 1;
    Node->lchild = Node->rchild = NULL;
    return Node;
}

//获取结点root的高度
int getHeight(node* root){
    if(root == NULL) return 0;
    return root->height;
}

//计算平衡因子
int getBalanceFactor(node* root){
    return getHeight(root->lchild) - getHeight(root->rchild);
}

//结点root所在子树的height等于其左子树的height与右子树的height的较大值加1
void updateHeight(node* root){
    root->height = max(getHeight(root->lchild), getHeight(root->rchild);
}

二、平衡二叉树的基本操作

1. 查找操作

• 查找方法和二叉查找树一致

void search(node* root, int x){
    if(root == NULL){
        printf("search failed\n");
        return;
    }
    if(x == root->data){
        printf("%d\n". root->data);
    }else if(x < root->data){
        search(root->lchild, x);
    }else{
        search(root->rchild, x);
    }
}

2. 插入操作

• 左旋(Left Rotation)

void L(node* &root){
    node* temp = root->rchild;
    root->rchild = temp->lchild;//步骤一
    temp->lchild = root;//步骤二
    updateHeight(root);//更新结点高度
    updateHeight(temp);
    root = temp;//步骤三
}

• 右旋(Right Rotation)

void R(node* &root){
    node* temp = root->lchild;
    root->lchild = temp->rchild;
    temp->rchild = root;
    updateHeight(root);
    updateHeight(temp);
    root = temp;
}

• LL:对root进行右旋,BF(root)=2,BF(root->lchild)=1
• LR:先对root->lchild进行左旋，再对root进行右旋。BF(root)=2, BF(root->lchild)=-1
• RR:对root进行左旋BF(root)=-2, BF(root->rchild)=-1
• RL:先对root->rchild进行右旋，再对root进行左旋,BF(root)=-2,BF(root->rchild)=1

//不考虑平衡的二叉排序树的插入操作
void insert(node* &root, int v){
    if(root == NULL){
        root = newNode(v);
        return;
    }
    if(v < root->v){
        insert(root->lchild, v);
    }else{
        insert(root->rchild, v);
    }
}

void insert(node* &root, int v){
    if(root == NULL){
        root = newNode(v);
        return;
    }
    if(v < root->v){
        insert(root->lchild, v);
        updateHeight(root);
        if(getBalanceFactor(root) == 2){
            if(getBalanceFactor(root->lchild) == 1){
                R(root);
            }else if(getBalanceFactor(root->lchild) == -1){
                L(root->lchild);
                R(root);
            }
        }else{
            insert(root->rchild, v);
            updateHeight(root);
            if(getBalanceFactor(root) == -2){
                if(getBalanceFactor(root-rchild) == -1){
                    L(root);
                }else if(getBalanceFactor(root-rchild) == 1）{
                    R(root->rchild);
                    L(root);
                }
            }
        }
    }
}

3. AVL树的建立

node* Create(int data[], int n){
    node* root = NULL;
    for(int i = 0; i < n; i++){
        insert(root, data[i]);
    }
    return root;
}