一、理解二叉搜索树
1.1 二叉搜索树的概念
- 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；
- 它的左右子树也分别为二叉搜索树；
- 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
如下图所示就是两个搜索二叉树
1.2 核心特性
1.2.1 多元化的结构： 灵活的数据结构
BST支持动态数据集合的高效操作，适合频繁插入、删除和查找的场景
1.2.2 天然的搜索优势：擅长搜索的数据结构
利用二叉树的分支特性，BST在平均情况下能实现O(logn)的搜索效率
二、二叉搜索树性能分析
2.1 时间复杂度分析
- 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：logN；
- 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；
注意：时间复杂度指的是最差情况下，所以时间复杂度为O(N)
最差情况如图：
那么这样的效率显然是无法满足我们需求的，因此后面博主会介绍二叉搜索树的变形——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
2.2 二分查找的局限性
另外需要说明的是，二分查找也可以实现O(logN)级别的查找效率，但是二分查找有两大缺陷
- 需要存储在支持下标随机访问的结构中，并且有序
- 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
如右图所示：数据越有序，插入结果越坏——高度高、递归深、效率低如左图所示：插入越无序的数据，左右会平衡一点，结果反而越好
三、实现二叉搜索树的定义
3.1 命名规范
二叉搜索树常简写为BST，提高代码可读性（SBT不好听），二叉搜索树也叫搜索二叉树
3.2 定义节点
代码语言：javascript
AI代码解释
template<class K> struct BSTreeNode { BSTreeNode<K>* _left; BSTreeNode<K>* _right; K _key; BSTreeNode(const K& key) :_left(nullptr) , _right(nullptr) , _key(key) { } };3.3 实践：完整的类定义
提供插入、查找、删除等基本操作的接口设计如下：
四、二叉树搜索的插入操作详解
4.1 插入算法流程
从根节点开始，根据键值大小选择左子树或右子树，直到找到空位置插入新节点。
插入分成以下三种情况：
- 树为空，则直接新增结点，赋值给root指针
- 树不空，按二叉搜索树性质，插入值比当前结点大就往右走，插入值比当前结点小就往左走，找到空位置，插入新结点
- 如果支持插入相等的值，插入值跟当前结点相等的值可以往右走，也可以往左走，找到空位置，插入新结点（要注意保持逻辑一致性，插入相等的值不要一会往右走，一会又往左走）
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