【2020-MOOC-浙江大学-陈越、何钦铭-数据结构】树（第四周的笔记和编程作业）

Love The Way You Lie 2023-07-17 06:11 1阅读 0赞

### 文章目录 ###

*  〇、前言
 *  一、二叉搜索树
 *  二、平衡二叉树
 *  三、平衡二叉树的调整
 *  四、是否同一棵二叉搜索树
 *  五、课后题
 *   *   *  1、04-树4 是否同一棵二叉搜索树 (25分)
         *  2、04-树5 Root of AVL Tree (25分)
         *  3、04-树6 Complete Binary Search Tree (30分)
         *  4、04-树7 二叉搜索树的操作集 (30分)
 *  总结

# 〇、前言 #

这两周开始跟着【MOOC-浙江大学-陈越、何钦铭-数据结构】进行数据结构与算法的学习，特此记录复习一下，虽然记不住，但是一直记一直记一直记，成为复读机就好了。  
![在这里插入图片描述][20200314162146148.png_pic_center]

# 一、二叉搜索树 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 1]

# 二、平衡二叉树 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 2]

# 三、平衡二叉树的调整 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 3]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 4]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 5]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 6]

# 四、是否同一棵二叉搜索树 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 7]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 8]  
实现在后面的第一题！

# 五、课后题 #

![在这里插入图片描述][2020031922004532.png]

--------------------

### 1、04-树4 是否同一棵二叉搜索树 (25分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 9]  
**输入样例:**

4 2  
3 1 4 2  
3 4 1 2  
3 2 4 1  
2 1  
2 1  
1 2  
0

**输出样例:**

Yes  
No  
No

--------------------

#include <stdio.h>
    #include <stdlib.h>
    
    typedef struct TreeNode *Tree;
    struct TreeNode { 
    	int v;
    	Tree Left, Right;
    	int flag;
    };
    
    Tree NewNode( int V )
    {  
    	Tree T = (Tree)malloc(sizeof(struct TreeNode));
    	T->v = V;
    	T->Left = T->Right = NULL;
    	T->flag = 0;
    	return T;
    }
    
    Tree Insert( Tree T, int V )
    { 
    	if ( !T ) 
    		T = NewNode(V);
    	else { 
    		if ( V>T->v )
    			T->Right = Insert( T->Right, V );
    		else
    			T->Left = Insert( T->Left, V );
    	}
    	return T;
    }
    
    Tree MakeTree( int N )
    {  
    	Tree T;
    	int i, V;
    	scanf("%d", &V);
    	T = NewNode(V);
    	for (i=1; i<N; i++) { 
    		scanf("%d", &V);
    		T = Insert(T, V);
    	}
    	return T;
    }
    
    int check ( Tree T, int V )
    { 
    	if ( T->flag ) { 
    		if ( V<T->v ) 
    			return check(T->Left, V);
    		else if ( V>T->v ) 
    			return check(T->Right, V);
    		else 
    			return 0;
    	}
    	else { 
    		if ( V==T->v ) { 
    			T->flag = 1;
    			return 1;
    		}
    		else 
    			return 0;
    	}
    }
    
    int Judge( Tree T, int N )
    { 
    	int i, V, flag = 0;
    	/* flag: 0代表目前还一致，1代表已经不一致*/
    	scanf("%d", &V);
    	if ( V!=T->v ) 
    		flag = 1;
    	else 
    		T->flag = 1;
    	for (i=1; i<N; i++) { 
    		scanf("%d", &V);
    		if ( (!flag) && (!check(T, V)) ) 
    			flag = 1;
    	}
    	if (flag) 
    		return 0;
    	else 
    		return 1;
    }
    
    void ResetT ( Tree T ) /* 清除T中各结点的flag标记 */
    { 
    	if (T->Left) ResetT(T->Left);
    	if (T->Right) ResetT(T->Right);
    	T->flag = 0;
    }
    
    void FreeTree ( Tree T ) /* 释放T的空间 */
    { 
    	if (T->Left) FreeTree(T->Left);
    	if (T->Right) FreeTree(T->Right);
    	free(T);
    }
    
    int main()
    { 
    	int N, L, i;
    	Tree T;
    	scanf("%d", &N);
    	while (N) { 
    		scanf("%d", &L);
    		T = MakeTree(N);
    		for (i=0; i<L; i++) { 
    			if (Judge(T, N)) 
    				printf("Yes\n");
    			else 
    				printf("No\n");
    			ResetT(T); /*清除T中的标记flag*/
    		}
    		FreeTree(T);
    		scanf("%d", &N);
    	}
    	return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 10]

--------------------

### 2、04-树5 Root of AVL Tree (25分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 11]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 12]  
**Sample Input 1:**

5  
88 70 61 96 120

**Sample Output 1:**

**Sample Input 2:**

7  
88 70 61 96 120 90 65

**Sample Output 2:**

--------------------

#include <stdio.h>
    #include <stdlib.h>
    #include <unistd.h>
    
    typedef int ElementType;
    typedef struct AVLNode *Position;
    typedef Position AVLTree; /* AVL树类型 */
    struct AVLNode{ 
        ElementType Data; /* 结点数据 */
        AVLTree Left;     /* 指向左子树 */
        AVLTree Right;    /* 指向右子树 */
        int Height;       /* 树高 */
    };
    
    int Max ( int a, int b )
    { 
        return a > b ? a : b;
    }
    
    int GetHeight(AVLTree A)
    { 
    	int MaxH, HR, HL;
    	if(A) { 
    		HL = GetHeight(A->Left);
    		HR = GetHeight(A->Right);
    		MaxH = (HL>HR)?HL:HR;
    		return MaxH+1;
    	}
    	return -1;
    }
    
    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;
        A->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);
    }
     
    AVLTree Insert( AVLTree T, ElementType X )
    {  /* 将X插入AVL树T中，并且返回调整后的AVL树 */
        if ( !T ) {  /* 若插入空树，则新建包含一个结点的树 */
            T = (AVLTree)malloc(sizeof(struct AVLNode));
            T->Data = X;
            T->Height = 0;
            T->Left = T->Right = NULL;
        } /* if (插入空树) 结束 */
     
        else if ( X < T->Data ) { 
            /* 插入T的左子树 */
            T->Left = Insert( T->Left, X);
            /* 如果需要左旋 */
            if ( GetHeight(T->Left)-GetHeight(T->Right) == 2 )
                if ( X < T->Left->Data ) 
                   T = SingleLeftRotation(T);      /* 左单旋 */
                else 
                   T = DoubleLeftRightRotation(T); /* 左-右双旋 */
        } /* else if (插入左子树) 结束 */
         
        else if ( X > T->Data ) { 
            /* 插入T的右子树 */
            T->Right = Insert( T->Right, X );
            /* 如果需要右旋 */
            if ( GetHeight(T->Left)-GetHeight(T->Right) == -2 )
                if ( X > T->Right->Data ) 
                   T = SingleRightRotation(T);     /* 右单旋 */
                else 
                   T = DoubleRightLeftRotation(T); /* 右-左双旋 */
        } /* else if (插入右子树) 结束 */
     
        /* else X == T->Data，无须插入 */
     
        /* 别忘了更新树高 */
        T->Height = Max( GetHeight(T->Left), GetHeight(T->Right) ) + 1;
         
        return T;
    }
    
    int main()
    { 
    	int N, i;
    	ElementType t;
    	AVLTree T=NULL;
    	scanf("%d", &N);
    	for (i=0; i<N; i++) { 
    	 	scanf("%d",&t);
    		T= Insert(T,t);
    	}
    	if(T)
    		printf("%d", T->Data);
    	return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 13]

--------------------

### 3、04-树6 Complete Binary Search Tree (30分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 14]  
**Sample Input:**

10  
1 2 3 4 5 6 7 8 9 0

**Sample Output:**

6 3 8 1 5 7 9 0 2 4

--------------------

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 15]

#include <stdlib.h>
    #include <stdio.h>
    #include <math.h>
    int arr[1000], rearr[1000];
    int compare( const void* a, const void* b ){ 
    	return *(int*)a - *(int*)b;
    }
    //利用二叉树的性质：
    //满二叉树第i层有 2^(i-1) 个结点， 
    //高为h的满二叉树有 2^h - 1 个结点（从1开始）
    int getLeftLength(int n){ 
    	double h, x, L, t;
    	h = (double)(int)( log((double)n+1) / log(2.0) );
    	//h = floor( log((double)n+1) / log(2.0) ); 
    	x = n - pow(2.0, h) + 1 ;
    	t = pow(2.0, h - 1.0);
    	x = x < t ? x : t;
    	L = t - 1 + x;
    	return (int)L;
    }
    void solve( int left, int right, int root ){ 
    	//初始调用： solve(0, n-1, 0);
    	int n, L, leftRoot, rightRoot;
    	n = right - left + 1;			//数组中的总个数
    	if(n == 0) return ;				//递归退出的条件
    	L = getLeftLength(n);			//计算出左子树的结点
    	rearr[root] = arr[left + L];	//将新的根结点放入新的数组
    	leftRoot = root * 2 + 1;		//左孩子
    	rightRoot = leftRoot + 1;		//右孩子
    	solve(left, left + L - 1, leftRoot);
    	solve(left + L + 1, right, rightRoot);
    }
    int main(){ 
    	int n;
    	scanf("%d", &n);
    	for(int i = 0; i < n; i++){ 
    		scanf("%d", &arr[i]);
    	}
    	qsort(arr, n, sizeof(int), compare);
    	solve(0, n-1, 0);
    	for(int i = 0; i < n; i++){ 
    		if( i != 0 ) printf(" ");
    		printf("%d", rearr[i]);
    	}
    	system("pause");
    	return 0;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 16]

--------------------

### 4、04-树7 二叉搜索树的操作集 (30分) ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 17]  
![在这里插入图片描述][20200319220734480.png]

#include <stdio.h>
    #include <stdlib.h>
    
    typedef int ElementType;
    typedef struct TNode *Position;
    typedef Position BinTree;
    struct TNode{ 
        ElementType Data;
        BinTree Left;
        BinTree Right;
    };
    
    void PreorderTraversal( BinTree BT ); /* 先序遍历，由裁判实现，细节不表 */
    void InorderTraversal( BinTree BT );  /* 中序遍历，由裁判实现，细节不表 */
    
    BinTree Insert( BinTree BST, ElementType X );
    BinTree Delete( BinTree BST, ElementType X );
    Position Find( BinTree BST, ElementType X );
    Position FindMin( BinTree BST );
    Position FindMax( BinTree BST );
    
    int main()
    { 
        BinTree BST, MinP, MaxP, Tmp;
        ElementType X;
        int N, i;
    
        BST = NULL;
        scanf("%d", &N);
        for ( i=0; i<N; i++ ) { 
            scanf("%d", &X);
            BST = Insert(BST, X);
        }
        printf("Preorder:"); PreorderTraversal(BST); printf("\n");
        MinP = FindMin(BST);
        MaxP = FindMax(BST);
        scanf("%d", &N);
        for( i=0; i<N; i++ ) { 
            scanf("%d", &X);
            Tmp = Find(BST, X);
            if (Tmp == NULL) printf("%d is not found\n", X);
            else { 
                printf("%d is found\n", Tmp->Data);
                if (Tmp==MinP) printf("%d is the smallest key\n", Tmp->Data);
                if (Tmp==MaxP) printf("%d is the largest key\n", Tmp->Data);
            }
        }
        scanf("%d", &N);
        for( i=0; i<N; i++ ) { 
            scanf("%d", &X);
            BST = Delete(BST, X);
        }
        printf("Inorder:"); InorderTraversal(BST); printf("\n");
    
        return 0;
    }
    /* 你的代码将被嵌在这里 */

**输入样例：**

10  
5 8 6 2 4 1 0 10 9 7  
5  
6 3 10 0 5  
5  
5 7 0 10 3

**输出样例：**

Preorder: 5 2 1 0 4 8 6 7 10 9  
6 is found  
3 is not found  
10 is found  
10 is the largest key  
0 is found  
0 is the smallest key  
5 is found  
Not Found  
Inorder: 1 2 4 6 8 9

--------------------

BinTree Insert( BinTree BST, ElementType X ){ 
    	if( !BST ){ 
    		BST = (BinTree)malloc(sizeof(struct TNode));
    		BST->Data = X;
    		BST->Left = BST->Right = NULL;
    	}
    	else { 
    		if( X < BST->Data ) BST->Left = Insert( BST->Left, X );
    		else if( X > BST->Data ) BST->Right = Insert( BST->Right, X );
    		//else if(X = BST->Data) do nothing
    	}
    	return BST;
    }
    BinTree Delete( BinTree BST, ElementType X ){ 
    	Position TMP;
    	if( !BST ) printf("Not Found\n");
    	else { 
    		if( X < BST->Data ) BST->Left = Delete( BST->Left, X );  //从左子树递归删除
    		else if( X > BST->Data ) BST->Right = Delete( BST->Right, X );  //从右子树递归删除
    		else {  //BST就是要删除的结点
    			if( BST->Left && BST->Right ){   //如果BST左右孩子都有
    				TMP = FindMin( BST->Right ); //从右子树中找到最小的结点来代替该结点
    				BST->Data = TMP->Data;
    				BST->Right = Delete( BST->Right, BST->Data );  //从右子树中把最小的结点删除
    			}
    			else {  
    				TMP = BST;
    				if( !BST->Left )	//如果只有右结点，或者没有结点
    					BST = BST->Right;
    				else				//只有左结点
    					BST = BST->Left;
    				free( TMP );
    			}
    		}
    	}
    	return BST;
    }
    Position Find( BinTree BST, ElementType X ){ 
    	if( !BST )	return NULL;
    	else if( X == BST->Data ) return BST;
    	else if( X > BST->Data ) return Find( BST->Right, X );
    	else if( X < BST->Data ) return Find( BST->Left, X );
    	return NULL;
    }
    //递归查找最小元素
    Position FindMin( BinTree BST ){ 
    	if( !BST ) return NULL;
    	else if( !BST->Left ) return BST;
    	else if( BST->Left ) FindMin( BST->Left );
    }
    //非递归查找最大元素
    Position FindMax( BinTree BST ){ 
    	if( BST )
    		while( BST->Right ) BST = BST->Right;
    	return BST;
    }

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70 18]

# 总结 #

简单总结下这周的学习内容，继续二叉树，我觉得自己非常有做调包侠的前途！！！

***如果想要更多的资源，欢迎关注 @我是管小亮，文字强迫症MAX~***

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![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1RlRnVpcm5ldmVy_size_16_color_FFFFFF_t_70_pic_center]

[20200314162146148.png_pic_center]: /images/20230528/900c7870030a4106a9d559a76a83b24c.png
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