C++：二叉搜索树

叉搜索树的性能分析和概念

1.1⼆叉搜索树的概念

⼆叉搜索树⼜称⼆叉排序树，它或者是⼀棵空树，是一种具有以下性质的⼆叉树:

•若它的左⼦树不为空，则左⼦树上所有结点的值都⼩于等于根结点的值•若它的右⼦树不为空，则右⼦树上所有结点的值都⼤于等于根结点的值• 它的左右⼦树都是⼆叉搜索树•⼆叉搜索树中可以⽀持插⼊相等的值，也可以不⽀持插⼊相等的值，具体看使⽤场景定义，例如： map/set/multimap/multiset系列底层就是⼆叉搜索树，其中map/set不⽀持插⼊相等 值，multimap/multiset⽀持插⼊相等值

1.2⼆叉搜索树的性能分析

由于它本身自己和它的子树都是⼆叉搜索树，这里观察上面的特点，不难发现：⼆叉搜索树的中序遍历一定是升序序列(所以⼆叉搜索树⼜称⼆叉排序树）。

⼆叉搜索树顾名思义，其只有由于搜索(或者搜索效率高)，下面我们来分析其最优情况和最插情况

最优情况下，⼆叉搜索树为完全⼆叉树(或者接近完全⼆叉树)，其⾼度为： log2N

最差情况下，⼆叉搜索树退化为单⽀树(或者类似单⽀)，其⾼度为：N

所以综合⽽⾔⼆叉搜索树增删查改时间复杂度为：O(N)

那么这样的效率显然是⽆法满⾜我们需求的，必须将二叉搜索树优化即⼆叉搜索树的变形：平衡⼆ 叉搜索树AVL树和红⿊树，才能适⽤于我们在内存中存储和搜索数据。

那有的人说使用另⼆分查找也可以实现O(log2N) 级别的查找效率为啥还有将二叉搜索树变形外。需要说明的是，⼆分查找也可以实现O(log 2N) 级别的查找效率，但是⼆分查找有两⼤缺陷：

1. 需要存储在⽀持下标随机访问的结构中，并且有序。

2. 插⼊和删除数据效率很低，因为存储在下标随机访问的结构中，插⼊和删除数据⼀般需要挪动数

据。

这⾥也就体现出了平衡⼆叉搜索树的价值。

二 ⼆叉搜索树增删查的实现，这里不支持修改(修改一个就可能不是二叉搜索树的结构了）

2.1⼆叉搜索树基本结构的实现

2.1.1节点的定义(由于后面实现⼆叉搜索树要访问节点成员所以这里使用struct定义(默认是公有))

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