04-树5 Root of AVL Tree

An AVL tree is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Figures 1-4 illustrate the rotation rules.

Now given a sequence of insertions, you are supposed to tell the root of the resulting AVL tree.
Input Specification:

Each input file contains one test case. For each case, the first line contains a positive integer NN (\le 20≤20) which is the total number of keys to be inserted. Then NN distinct integer keys are given in the next line. All the numbers in a line are separated by a space.

Output Specification:

For each test case, print the root of the resulting AVL tree in one line.

Sample Input 1:

5
88 70 61 96 120
Sample Output 1:

70
Sample Input 2:

7
88 70 61 96 120 90 65
Sample Output 2:

88

#include<stdio.h>
#include<stdlib.h>
#include<stdlib.h>
typedef struct AVLNode *Position;
typedef Position AVLTree;
typedef struct AVLNode{
    int data;
    AVLTree Left,Right;
    int Height;
};
AVLTree NewNode(int V); 
int Max(int a,int b);
AVLTree Insert(AVLTree T,int V);
AVLTree SingleLeftRotation(AVLTree A);
AVLTree SingleRightRotation (AVLTree A);
AVLTree DoubleLeftRightRotation(AVLTree A);
AVLTree DoubleRightLeftRotation(AVLTree A);
int GetHeight(AVLTree A);
int main(){
    int N,V,i;
    AVLTree T;
    scanf("%d",&N);
    if(N>=1&&N<=20){
        scanf("%d",&V);
        T=NewNode(V);
        for(i=1;i<N;i++){
            scanf("%d",&V);
            T=Insert(T,V);
        }
        printf("%d\n",T->data);
    }
    return 0;
}
AVLTree NewNode(int V){
    AVLTree T;
    T=(AVLTree)malloc(sizeof(struct AVLNode));
    T->data = V;
    T->Left= T->Right = NULL;
    T->Height=0;
    return T; 
}
int Max(int a,int b){
    return a>b?a:b; 
}
AVLTree Insert(AVLTree T,int V){
    if(!T){
        T = NewNode(V);
    }
    else if(V<T->data){
        T->Left = Insert(T->Left,V);
        if(GetHeight(T->Left)-GetHeight(T->Right)==2){
            if(V<T->Left->data)
            T = SingleLeftRotation(T);
            else
            T = DoubleLeftRightRotation(T);
        }
    }
    else if(V>T->data){
        T->Right = Insert(T->Right,V);
        if(GetHeight(T->Left)-GetHeight(T->Right)==-2){
            if(V>T->Right->data)
            T= SingleRightRotation(T);
            else
            T= DoubleRightLeftRotation(T);
        }
    }
    T->Height = Max(GetHeight(T->Left),GetHeight(T->Right))+1;
    return T;
}
AVLTree SingleLeftRotation(AVLTree A){
    AVLTree B=A->Left;
    A->Left=B->Right;
    B->Right=A;
    A->Height=Max(GetHeight(A->Left),GetHeight(A->Right))+1;
    B->Height=Max(GetHeight(B->Left),A->Height)+1;
    return B; 
}
AVLTree SingleRightRotation (AVLTree A){
    AVLTree B=A->Right;
    A->Right = B->Left;
    B->Left = A;
    A->Height=Max(GetHeight(A->Left),GetHeight(A->Right))+1;
    B->Height=Max(GetHeight(B->Right),A->Height)+1;
    return B;
}
AVLTree DoubleLeftRightRotation(AVLTree A){
    A->Left = SingleRightRotation(A->Left);
    return SingleLeftRotation(A); 
} 
AVLTree DoubleRightLeftRotation(AVLTree A){
    A->Right = SingleLeftRotation(A->Right);
    return SingleRightRotation(A);
}
int GetHeight(AVLTree A){
    int depth1,depth2;
    if(!A){
        return 0;
    }
    else{

            depth1 = GetHeight(A->Left);
            depth2 = GetHeight(A->Right);
    }
    return depth1>depth2?depth1+1:depth2+1;
}
以下是使用 C 语言根据输入的插入序列构建 AVL 并输出根节点的代码: ```c #include <stdio.h> #include <stdlib.h> // 定义 AVL 节点结构 typedef struct AVLNode { int key; struct AVLNode *left; struct AVLNode *right; int height; } AVLNode; // 获取节点高度 int height(AVLNode *N) { if (N == NULL) return 0; return N->height; } // 获取两个整数中的最大值 int max(int a, int b) { return (a > b)? a : b; } // 创建一个新节点 AVLNode* newNode(int key) { AVLNode* node = (AVLNode*)malloc(sizeof(AVLNode)); node->key = key; node->left = NULL; node->right = NULL; node->height = 1; return(node); } // 右旋操作 AVLNode *rightRotate(AVLNode *y) { AVLNode *x = y->left; AVLNode *T2 = x->right; x->right = y; y->left = T2; y->height = max(height(y->left), height(y->right)) + 1; x->height = max(height(x->left), height(x->right)) + 1; return x; } // 左旋操作 AVLNode *leftRotate(AVLNode *x) { AVLNode *y = x->right; AVLNode *T2 = y->left; y->left = x; x->right = T2; x->height = max(height(x->left), height(x->right)) + 1; y->height = max(height(y->left), height(y->right)) + 1; return y; } // 获取平衡因子 int getBalance(AVLNode *N) { if (N == NULL) return 0; return height(N->left) - height(N->right); } // 插入节点 AVLNode* insert(AVLNode* node, int key) { if (node == NULL) return(newNode(key)); if (key < node->key) node->left = insert(node->left, key); else if (key > node->key) node->right = insert(node->right, key); else return node; node->height = 1 + max(height(node->left), height(node->right)); int balance = getBalance(node); // 左左情况 if (balance > 1 && key < node->left->key) return rightRotate(node); // 右右情况 if (balance < -1 && key > node->right->key) return leftRotate(node); // 左右情况 if (balance > 1 && key > node->left->key) { node->left = leftRotate(node->left); return rightRotate(node); } // 右左情况 if (balance < -1 && key < node->right->key) { node->right = rightRotate(node->right); return leftRotate(node); } return node; } int main() { AVLNode *root = NULL; int values[] = {2, 1, 4, 5, 9, 3, 6, 7}; int n = sizeof(values) / sizeof(values[0]); for (int i = 0; i < n; i++) { root = insert(root, values[i]); } if (root != NULL) { printf("最终 AVL 的根节点值为: %d\n", root->key); } else { printf("AVL 。\n"); } return 0; } ``` ### 代码解释 1. **AVL 节点结构**:定义了 `AVLNode` 结构体,包含节点的值、左右子节点指针和高度信息。 2. **插入节点**:使用 `insert` 函数将节点插入AVL 中,并在插入后进行平衡调整。 3. **平衡调整**:通过 `rightRotate` 和 `leftRotate` 函数进行右旋和左旋操作,以保持的平衡。 4. **主函数**:创建一个初始AVL ,然后将输入序列中的元素依次插入中,最后输出根节点的值。
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