数据结构笔记——二叉排序树

男娘i 2023-02-17 02:15 8阅读 0赞

**目录**

一、二叉排序树的定义

二、二叉排序树的查找

三、二叉树的插入

四、二叉排序树的构造

五、二叉排序树的删除

六、查找效率分析

七、总结

## **一、二叉排序树的定义** ##

二叉排序树，又称二叉查找树（BST），一棵二叉树或者是空二叉树，或者是具有如下性质的二叉树：

左子树上所有结点的关键字均小于根节点的关键字；

右子树上所有结点的关键字均大于根节点的关键字。

左子树和右子树又各是一棵二叉排序树。

注：

①左子树结点值<根结点值<右子树结点值。

②进行中序遍历，可以得到一个递增的有序序列。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70][]![20200610184814509.png][]

## **二、二叉排序树的查找** ##

若树非空，目标值与根节点的值比较：

若相等，则查找成功；

若小于根节点，则在左子树上查找，否则在右子树上查找。

查找成功，返回结点指针；查找失败返回NULL

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 1][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 2][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 3][]

**非递归实现，最坏空间复杂度O(1)**

typedef struct BSTNode{
        int key;
        struct BSTNode *lchild,rchild;
    }BSTNode,*BSTree;
    
    BSTNode *BST_Search(BSTree T,int key){
        while(T != NULL && key != T->key){
            if( key < T->key)
                T = T->lchild;
            else
                T = T->rchild;
        }
        return T;
    }

**递归实现，最坏空间复杂度O(h)**

BSTNode *BSTSearch(BSTree T,int key){
        if(T == NULL)
            return NULL;
        if(key == T->key)
            return T;
        else if(key < T->key)
            return BSTSearch(T->lchild,key);
        else
            return BSTSearch(T->rchild,key);
    }

## **三、二叉树的插入** ##

若原二叉排序树为空，则直接插入结点；否则，若关键字k小于根节点值，则插入到左子树，若关键字k大于根结点值，则插入到右子

int BST_Insert(BSTree &T,int k){
        if(T == NULL){
            T = (BSTree)malloc(sizeof(BSTNode));
            T->key = k;
            T->lchild = T->rchild = NULL;
            return 1;
        }
        else if(k == T->key)
            return 0;
        else if(k < T->key)
            return BST_Insert(T->lchild,k);
        else
            return BST_Insert(T->rchild,k);
    }

## **四、二叉排序树的构造** ##

void Creat_BST(BSTree &T,int str[],int n){
        T = NULL;
        int i = 0;
        while(i < n){
            BST_Insert(T,str[i]);
            i++;
        }
    }

例1：按照序列str=\[50,66,60,26,21,30,70,68\]建立BST

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 4][]

例2：按照序列str=\[50,26,21,30,66,60,70,68\]建立BST

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 4][]

注：不同的关键字序列可能得到同款二叉排序树

例3：按照序列str=\[26,21,30,50,60,66,68,70\]建立BST

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 5][]

## **五、二叉排序树的删除** ##

先搜索找到目标结点：

**①若被删除结点z是叶子结点，则直接删除，不会破坏二叉排序树的性质。**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 6][]

**②若结点z只有一颗左子树或右子树，则让z的子树称为z父结点的子树，替代z的位置**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 7][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 8][]

**③若结点z有左、右两棵子树，则令z的直接后继（或直接前驱）替代z，然后从二叉排序树中删去这个直接后继（或直接前驱）**

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 9][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 10][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 11][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 12][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 13][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 14][]

## **六、查找效率分析** ##

查找长度——在查找运算中，需要对比关键字的次数称为查找长度，反映了查找操作时间复杂度

注：

①若树高h，找到最下层的一个结点需要对比h次

②最好情况：n个结点的二叉树最小高度为\[log2n\] + 1，平均查找长度 = O(log2n)

③最坏情况：每个结点只有一个分支，树高h = 结点数n，平均查找长度 = O(n)

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 15][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 5][]

**查找成功的平均查找长度ASL**

①ASL = （1\*1 + 2\*2 + 3\*4 + 4\*1） / 8 = 2.625

②ASL = （1\*1 + 2\*2 + 3\*1 + 4\*1 + 5\*1 + 6\*1 + 7\*1） / 8 = 3.75

**平衡二叉树：树上任一结点的左子树和右子树的深度之差不超过1。**

![20200610191722512.png][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 16][]

注：n个结点的二叉树最小高度为\[log2n\] + 1（完全二叉树），而平衡二叉树高度与完全二叉树同等数量级

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 17][]![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 18][]

**查找失败的平均查找长度ASL**

①ASL=(3\*7 + 4\*2) / 9 = 3.22

②ASL=(2\*3 + 3 + 4 + 5 + 6 + 7\*2) / 9 = 4.22

## **七、总结** ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQwMzEwMTQ4_size_16_color_FFFFFF_t_70 19][]

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