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# 树
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* [二叉搜索树](binary-search-tree)
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* [AVL树](avl-tree)
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* [红黑树](red-black-tree)
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* [线段树](segment-tree) - with min/max/sum range queries examples
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* [芬威克树/Fenwick Tree](fenwick-tree) (Binary Indexed Tree)
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- [二叉搜索树](binary-search-tree/README.zh-CN.md)
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- [AVL 树](avl-tree)
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- [红黑树](red-black-tree)
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- [线段树](segment-tree) - with min/max/sum range queries examples
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- [芬威克树/Fenwick Tree](fenwick-tree) (Binary Indexed Tree)
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在计算机科学中, **树(tree)** 是一种广泛使用的抽象数据类型(ADT)— 或实现此ADT的数据结构 — 模拟分层树结构, 具有根节点和有父节点的子树,表示为一组链接节点。
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在计算机科学中, **树(tree)** 是一种广泛使用的抽象数据类型(ADT)— 或实现此 ADT 的数据结构 — 模拟分层树结构, 具有根节点和有父节点的子树,表示为一组链接节点。
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树可以被(本地地)递归定义为一个(始于一个根节点的)节点集, 每个节点都是一个包含了值的数据结构, 除了值,还有该节点的节点引用列表(子节点)一起。
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树的节点之间没有引用重复的约束。
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一棵简单的无序树; 在下图中:
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标记为7的节点具有两个子节点, 标记为2和6;
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一个父节点,标记为2,作为根节点, 在顶部,没有父节点。
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标记为 7 的节点具有两个子节点, 标记为 2 和 6;
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一个父节点,标记为 2,作为根节点, 在顶部,没有父节点。
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*Made with [okso.app](https://okso.app)*
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_Made with [okso.app](https://okso.app)_
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## 参考
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- [Wikipedia](https://en.wikipedia.org/wiki/Tree_(data_structure))
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- [Wikipedia](<https://en.wikipedia.org/wiki/Tree_(data_structure)>)
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- [YouTube](https://www.youtube.com/watch?v=oSWTXtMglKE&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=8)
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@ -1,7 +1,7 @@
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# Binary Search Tree
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_Read this in other languages:_
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[_Português_](README.pt-BR.md)
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[_Português_](README.pt-BR.md),[_简体中文_](README.zh-CN.md)
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In computer science, **binary search trees** (BST), sometimes called
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ordered or sorted binary trees, are a particular type of container:
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@ -30,7 +30,7 @@ The leaves are not drawn.
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*Made with [okso.app](https://okso.app)*
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_Made with [okso.app](https://okso.app)_
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## Pseudocode for Basic Operations
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@ -87,7 +87,6 @@ contains(root, value)
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end contains
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```
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### Deletion
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```text
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@ -265,7 +264,7 @@ end postorder
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### Time Complexity
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| Access | Search | Insertion | Deletion |
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| Access | Search | Insertion | Deletion |
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| :-------: | :-------: | :-------: | :-------: |
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| O(log(n)) | O(log(n)) | O(log(n)) | O(log(n)) |
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268
src/data-structures/tree/binary-search-tree/README.zh-CN.md
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268
src/data-structures/tree/binary-search-tree/README.zh-CN.md
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@ -0,0 +1,268 @@
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# 二叉搜索树
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_Read this in other languages:_
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[_Português_](README.pt-BR.md),[_English_](README.md)
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在计算机科学中, **二叉搜索树** (BST), 也称为有序二叉树或排序二叉树, 是一种特殊的容器:
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在内存中存储“元素”的数据结构（如数字，名称等）。二叉搜索树可以快速查找，添加和删除元素，也可用于
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构建动态元素集或在表中根据键查找元素值 （例：通过某人姓名找到某人的手机号）。
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二叉搜索树为有序序列，所以在进行查找或其他操作时可以使用二分查找原理：
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在树中寻找键（或插入新键的位置）时，查找过程为: 从根到叶遍历树，通过比较
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存储在树的节点中的键来判断继续向左或向右搜索子树。 平均而言，这意味着每次
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比较都允许跳过大约一半的操作，这样每个查找、插入或删除所花费的时间为
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树中存储的项目数的对数。 这是比按键在（未排序的）数组中查找元素所需的
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线性时间要好得多，但比在哈希表中相应的操作慢。
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下图为一个大小为 9，深度为 3，8 为根结点的二叉搜索树。
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叶子节点没有被绘制。
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_Made with [okso.app](https://okso.app)_
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## 基础操作的伪代码
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### 插入
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```text
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insert(value)
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Pre: value has passed custom type checks for type T
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Post: value has been placed in the correct location in the tree
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if root = ø
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root ← node(value)
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else
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insertNode(root, value)
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end if
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end insert
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```
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```text
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insertNode(current, value)
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Pre: current is the node to start from
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Post: value has been placed in the correct location in the tree
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if value < current.value
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if current.left = ø
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current.left ← node(value)
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else
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InsertNode(current.left, value)
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end if
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else
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if current.right = ø
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current.right ← node(value)
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else
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InsertNode(current.right, value)
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end if
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end if
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end insertNode
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```
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### 查找
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```text
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contains(root, value)
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Pre: root is the root node of the tree, value is what we would like to locate
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Post: value is either located or not
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if root = ø
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return false
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end if
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if root.value = value
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return true
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else if value < root.value
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return contains(root.left, value)
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else
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return contains(root.right, value)
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end if
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end contains
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```
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### 删除
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```text
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remove(value)
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Pre: value is the value of the node to remove, root is the node of the BST
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count is the number of items in the BST
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Post: node with value is removed if found in which case yields true, otherwise false
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nodeToRemove ← findNode(value)
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if nodeToRemove = ø
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return false
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end if
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parent ← findParent(value)
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if count = 1
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root ← ø
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else if nodeToRemove.left = ø and nodeToRemove.right = ø
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if nodeToRemove.value < parent.value
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parent.left ← nodeToRemove.right
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else
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parent.right ← nodeToRemove.right
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end if
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else if nodeToRemove.left != ø and nodeToRemove.right != ø
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next ← nodeToRemove.right
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while next.left != ø
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next ← next.left
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end while
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if next != nodeToRemove.right
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remove(next.value)
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nodeToRemove.value ← next.value
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else
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nodeToRemove.value ← next.value
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nodeToRemove.right ← nodeToRemove.right.right
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end if
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else
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if nodeToRemove.left = ø
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next ← nodeToRemove.right
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else
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next ← nodeToRemove.left
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end if
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if root = nodeToRemove
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root = next
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else if parent.left = nodeToRemove
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parent.left = next
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else if parent.right = nodeToRemove
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parent.right = next
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end if
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end if
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count ← count - 1
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return true
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end remove
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```
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### 查找某个节点的父节点
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```text
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findParent(value, root)
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Pre: value is the value of the node we want to find the parent of
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root is the root node of the BST and is != ø
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Post: a reference to the prent node of value if found; otherwise ø
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if value = root.value
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return ø
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end if
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if value < root.value
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if root.left = ø
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return ø
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else if root.left.value = value
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return root
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else
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return findParent(value, root.left)
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end if
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else
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if root.right = ø
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return ø
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else if root.right.value = value
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return root
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else
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return findParent(value, root.right)
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end if
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end if
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end findParent
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```
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### 查找节点
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```text
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findNode(root, value)
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Pre: value is the value of the node we want to find the parent of
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root is the root node of the BST
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Post: a reference to the node of value if found; otherwise ø
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if root = ø
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return ø
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end if
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if root.value = value
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return root
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else if value < root.value
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return findNode(root.left, value)
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else
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return findNode(root.right, value)
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end if
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end findNode
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```
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### 查找最小值
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```text
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findMin(root)
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Pre: root is the root node of the BST
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root = ø
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Post: the smallest value in the BST is located
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if root.left = ø
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return root.value
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end if
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findMin(root.left)
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end findMin
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```
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### 查找最大值
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```text
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findMax(root)
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Pre: root is the root node of the BST
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root = ø
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Post: the largest value in the BST is located
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if root.right = ø
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return root.value
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end if
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findMax(root.right)
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end findMax
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```
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### 遍历
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#### 中序遍历
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```text
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inorder(root)
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Pre: root is the root node of the BST
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Post: the nodes in the BST have been visited in inorder
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if root != ø
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inorder(root.left)
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yield root.value
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inorder(root.right)
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end if
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end inorder
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```
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#### 前序遍历
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```text
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preorder(root)
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Pre: root is the root node of the BST
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Post: the nodes in the BST have been visited in preorder
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if root != ø
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yield root.value
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preorder(root.left)
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preorder(root.right)
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end if
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end preorder
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```
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#### 后序遍历
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```text
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postorder(root)
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Pre: root is the root node of the BST
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Post: the nodes in the BST have been visited in postorder
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if root != ø
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postorder(root.left)
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postorder(root.right)
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yield root.value
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end if
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end postorder
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```
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## 复杂度
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### 时间复杂度
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| Access | Search | Insertion | Deletion |
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| :-------: | :-------: | :-------: | :-------: |
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| O(log(n)) | O(log(n)) | O(log(n)) | O(log(n)) |
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### 空间复杂度
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O(n)
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## 参考资料
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- [Wikipedia](https://en.wikipedia.org/wiki/Binary_search_tree)
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- [Inserting to BST on YouTube](https://www.youtube.com/watch?v=wcIRPqTR3Kc&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=9&t=0s)
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- [BST Interactive Visualisations](https://www.cs.usfca.edu/~galles/visualization/BST.html)
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