2023-02-16 03:39:01 +08:00
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# 二叉搜索树
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2022-11-22 17:47:26 +08:00
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「二叉搜索树 Binary Search Tree」满足以下条件:
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2023-04-09 04:32:17 +08:00
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1. 对于根节点,左子树中所有节点的值 $<$ 根节点的值 $<$ 右子树中所有节点的值;
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2023-04-10 23:59:22 +08:00
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2. 任意节点的左、右子树也是二叉搜索树,即同样满足条件 `1.` ;
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2022-11-22 17:47:26 +08:00
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2023-02-26 18:18:34 +08:00
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![二叉搜索树](binary_search_tree.assets/binary_search_tree.png)
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2022-11-22 17:47:26 +08:00
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2023-02-16 03:39:01 +08:00
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## 二叉搜索树的操作
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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### 查找节点
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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给定目标节点值 `num` ,可以根据二叉搜索树的性质来查找。我们声明一个节点 `cur` ,从二叉树的根节点 `root` 出发,循环比较节点值 `cur.val` 和 `num` 之间的大小关系
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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- 若 `cur.val < num` ,说明目标节点在 `cur` 的右子树中,因此执行 `cur = cur.right` ;
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- 若 `cur.val > num` ,说明目标节点在 `cur` 的左子树中,因此执行 `cur = cur.left` ;
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2023-04-10 23:59:22 +08:00
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- 若 `cur.val = num` ,说明找到目标节点,跳出循环并返回该节点;
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<1>"
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2023-04-10 23:59:22 +08:00
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![bst_search_step1](binary_search_tree.assets/bst_search_step1.png)
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<2>"
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2023-02-25 23:35:39 +08:00
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![bst_search_step2](binary_search_tree.assets/bst_search_step2.png)
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<3>"
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2023-02-25 23:35:39 +08:00
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![bst_search_step3](binary_search_tree.assets/bst_search_step3.png)
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<4>"
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2023-02-25 23:35:39 +08:00
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![bst_search_step4](binary_search_tree.assets/bst_search_step4.png)
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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二叉搜索树的查找操作与二分查找算法的工作原理一致,都是每轮排除一半情况。循环次数最多为二叉树的高度,当二叉树平衡时,使用 $O(\log n)$ 时间。
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2022-11-22 17:47:26 +08:00
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=== "Java"
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```java title="binary_search_tree.java"
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2023-02-07 04:43:52 +08:00
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[class]{BinarySearchTree}-[func]{search}
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2022-11-22 17:47:26 +08:00
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```
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2022-11-29 02:21:49 +08:00
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=== "C++"
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```cpp title="binary_search_tree.cpp"
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2023-02-08 04:17:26 +08:00
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[class]{BinarySearchTree}-[func]{search}
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2022-11-29 02:21:49 +08:00
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```
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=== "Python"
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```python title="binary_search_tree.py"
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2023-02-06 23:23:21 +08:00
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[class]{BinarySearchTree}-[func]{search}
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2022-11-29 02:21:49 +08:00
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```
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=== "Go"
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```go title="binary_search_tree.go"
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2023-02-09 04:45:06 +08:00
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[class]{binarySearchTree}-[func]{search}
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2022-11-29 02:21:49 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "JavaScript"
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2023-02-08 04:27:55 +08:00
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```javascript title="binary_search_tree.js"
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2023-02-08 19:45:06 +08:00
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[class]{}-[func]{search}
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2022-12-03 01:31:29 +08:00
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```
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=== "TypeScript"
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```typescript title="binary_search_tree.ts"
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2023-02-08 19:45:06 +08:00
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[class]{}-[func]{search}
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2022-12-03 01:31:29 +08:00
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```
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=== "C"
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```c title="binary_search_tree.c"
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2023-02-11 18:22:27 +08:00
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[class]{binarySearchTree}-[func]{search}
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2022-12-03 01:31:29 +08:00
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```
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=== "C#"
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```csharp title="binary_search_tree.cs"
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2023-02-08 22:18:02 +08:00
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[class]{BinarySearchTree}-[func]{search}
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2022-12-03 01:31:29 +08:00
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```
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2023-01-08 19:41:05 +08:00
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=== "Swift"
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```swift title="binary_search_tree.swift"
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2023-02-08 20:30:05 +08:00
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[class]{BinarySearchTree}-[func]{search}
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2023-01-08 19:41:05 +08:00
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```
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2023-02-01 22:03:04 +08:00
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=== "Zig"
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```zig title="binary_search_tree.zig"
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2023-02-09 22:57:25 +08:00
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[class]{BinarySearchTree}-[func]{search}
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2023-02-01 22:03:04 +08:00
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```
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2023-04-09 04:32:17 +08:00
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### 插入节点
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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给定一个待插入元素 `num` ,为了保持二叉搜索树“左子树 < 根节点 < 右子树”的性质,插入操作分为两步:
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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1. **查找插入位置**:与查找操作相似,从根节点出发,根据当前节点值和 `num` 的大小关系循环向下搜索,直到越过叶节点(遍历至 $\text{null}$ )时跳出循环;
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2. **在该位置插入节点**:初始化节点 `num` ,将该节点置于 $\text{null}$ 的位置;
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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二叉搜索树不允许存在重复节点,否则将违反其定义。因此,若待插入节点在树中已存在,则不执行插入,直接返回。
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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![在二叉搜索树中插入节点](binary_search_tree.assets/bst_insert.png)
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2022-11-22 17:47:26 +08:00
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=== "Java"
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```java title="binary_search_tree.java"
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2023-02-07 04:43:52 +08:00
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[class]{BinarySearchTree}-[func]{insert}
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2022-11-22 17:47:26 +08:00
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```
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2022-11-29 02:21:49 +08:00
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=== "C++"
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```cpp title="binary_search_tree.cpp"
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2023-02-08 04:17:26 +08:00
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[class]{BinarySearchTree}-[func]{insert}
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2022-11-29 02:21:49 +08:00
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```
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=== "Python"
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```python title="binary_search_tree.py"
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2023-02-06 23:23:21 +08:00
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[class]{BinarySearchTree}-[func]{insert}
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2022-11-29 02:21:49 +08:00
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```
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=== "Go"
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```go title="binary_search_tree.go"
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2023-02-09 04:45:06 +08:00
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[class]{binarySearchTree}-[func]{insert}
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2022-11-29 02:21:49 +08:00
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```
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2022-12-03 01:31:29 +08:00
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=== "JavaScript"
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2023-02-08 04:27:55 +08:00
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```javascript title="binary_search_tree.js"
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2023-02-08 19:45:06 +08:00
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[class]{}-[func]{insert}
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2022-12-03 01:31:29 +08:00
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```
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=== "TypeScript"
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```typescript title="binary_search_tree.ts"
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2023-02-08 19:45:06 +08:00
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[class]{}-[func]{insert}
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2022-12-03 01:31:29 +08:00
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```
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=== "C"
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```c title="binary_search_tree.c"
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2023-02-11 18:22:27 +08:00
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[class]{binarySearchTree}-[func]{insert}
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2022-12-03 01:31:29 +08:00
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```
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=== "C#"
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```csharp title="binary_search_tree.cs"
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2023-02-08 22:18:02 +08:00
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[class]{BinarySearchTree}-[func]{insert}
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2022-12-03 01:31:29 +08:00
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```
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2023-01-08 19:41:05 +08:00
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=== "Swift"
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```swift title="binary_search_tree.swift"
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2023-02-08 20:30:05 +08:00
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[class]{BinarySearchTree}-[func]{insert}
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2023-01-08 19:41:05 +08:00
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```
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2023-02-01 22:03:04 +08:00
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=== "Zig"
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```zig title="binary_search_tree.zig"
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2023-02-09 22:57:25 +08:00
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[class]{BinarySearchTree}-[func]{insert}
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2023-02-01 22:03:04 +08:00
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```
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2023-04-10 23:59:22 +08:00
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为了插入节点,我们需要利用辅助节点 `pre` 保存上一轮循环的节点,这样在遍历至 $\text{null}$ 时,我们可以获取到其父节点,从而完成节点插入操作。
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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与查找节点相同,插入节点使用 $O(\log n)$ 时间。
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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### 删除节点
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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与插入节点类似,我们需要在删除操作后维持二叉搜索树的“左子树 < 根节点 < 右子树”的性质。首先,我们需要在二叉树中执行查找操作,获取待删除节点。接下来,根据待删除节点的子节点数量,删除操作需分为三种情况:
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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当待删除节点的子节点数量 $= 0$ 时,表示待删除节点是叶节点,可以直接删除。
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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![在二叉搜索树中删除节点(度为 0)](binary_search_tree.assets/bst_remove_case1.png)
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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当待删除节点的子节点数量 $= 1$ 时,将待删除节点替换为其子节点即可。
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2022-11-22 17:47:26 +08:00
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2023-04-09 04:32:17 +08:00
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![在二叉搜索树中删除节点(度为 1)](binary_search_tree.assets/bst_remove_case2.png)
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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当待删除节点的子节点数量 $= 2$ 时,删除操作分为三步:
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2023-04-14 05:47:20 +08:00
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1. 找到待删除节点在“中序遍历序列”中的下一个节点,记为 `tmp` ;
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2. 在树中递归删除节点 `tmp` ;
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3. 用 `tmp` 的值覆盖待删除节点的值;
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<1>"
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2023-04-10 23:59:22 +08:00
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![bst_remove_case3_step1](binary_search_tree.assets/bst_remove_case3_step1.png)
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<2>"
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2023-02-25 23:35:39 +08:00
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![bst_remove_case3_step2](binary_search_tree.assets/bst_remove_case3_step2.png)
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<3>"
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2023-02-25 23:35:39 +08:00
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![bst_remove_case3_step3](binary_search_tree.assets/bst_remove_case3_step3.png)
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2022-11-22 17:47:26 +08:00
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2023-02-22 00:57:43 +08:00
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=== "<4>"
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2023-02-25 23:35:39 +08:00
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![bst_remove_case3_step4](binary_search_tree.assets/bst_remove_case3_step4.png)
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2022-11-22 17:47:26 +08:00
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2023-04-10 23:59:22 +08:00
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删除节点操作同样使用 $O(\log n)$ 时间,其中查找待删除节点需要 $O(\log n)$ 时间,获取中序遍历后继节点需要 $O(\log n)$ 时间。
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2022-11-22 17:47:26 +08:00
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=== "Java"
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```java title="binary_search_tree.java"
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2023-02-07 04:43:52 +08:00
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[class]{BinarySearchTree}-[func]{remove}
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2022-11-22 17:47:26 +08:00
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```
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2022-11-29 02:21:49 +08:00
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=== "C++"
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```cpp title="binary_search_tree.cpp"
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2023-02-08 04:17:26 +08:00
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[class]{BinarySearchTree}-[func]{remove}
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2022-11-29 02:21:49 +08:00
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```
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=== "Python"
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```python title="binary_search_tree.py"
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2023-02-06 23:23:21 +08:00
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[class]{BinarySearchTree}-[func]{remove}
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2022-11-29 02:21:49 +08:00
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```
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=== "Go"
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```go title="binary_search_tree.go"
|
2023-02-09 04:45:06 +08:00
|
|
|
|
[class]{binarySearchTree}-[func]{remove}
|
2022-11-29 02:21:49 +08:00
|
|
|
|
```
|
|
|
|
|
|
2022-12-03 01:31:29 +08:00
|
|
|
|
=== "JavaScript"
|
|
|
|
|
|
2023-02-08 04:27:55 +08:00
|
|
|
|
```javascript title="binary_search_tree.js"
|
2023-02-08 19:45:06 +08:00
|
|
|
|
[class]{}-[func]{remove}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "TypeScript"
|
|
|
|
|
|
|
|
|
|
```typescript title="binary_search_tree.ts"
|
2023-02-08 19:45:06 +08:00
|
|
|
|
[class]{}-[func]{remove}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
|
|
|
|
```c title="binary_search_tree.c"
|
2023-02-11 18:22:27 +08:00
|
|
|
|
[class]{binarySearchTree}-[func]{remove}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
|
|
|
|
```csharp title="binary_search_tree.cs"
|
2023-02-08 22:18:02 +08:00
|
|
|
|
[class]{BinarySearchTree}-[func]{remove}
|
2022-12-03 01:31:29 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-01-08 19:41:05 +08:00
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title="binary_search_tree.swift"
|
2023-02-08 20:30:05 +08:00
|
|
|
|
[class]{BinarySearchTree}-[func]{remove}
|
2023-01-08 19:41:05 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-02-01 22:03:04 +08:00
|
|
|
|
=== "Zig"
|
|
|
|
|
|
|
|
|
|
```zig title="binary_search_tree.zig"
|
2023-02-09 22:57:25 +08:00
|
|
|
|
[class]{BinarySearchTree}-[func]{remove}
|
2023-02-01 22:03:04 +08:00
|
|
|
|
```
|
|
|
|
|
|
2023-02-06 19:57:19 +08:00
|
|
|
|
### 排序
|
|
|
|
|
|
2023-04-10 23:59:22 +08:00
|
|
|
|
我们知道,二叉树的中序遍历遵循“左 $\rightarrow$ 根 $\rightarrow$ 右”的遍历顺序,而二叉搜索树满足“左子节点 $<$ 根节点 $<$ 右子节点”的大小关系。因此,在二叉搜索树中进行中序遍历时,总是会优先遍历下一个最小节点,从而得出一个重要性质:**二叉搜索树的中序遍历序列是升序的**。
|
2023-02-06 19:57:19 +08:00
|
|
|
|
|
2023-04-10 23:59:22 +08:00
|
|
|
|
利用中序遍历升序的性质,我们在二叉搜索树中获取有序数据仅需 $O(n)$ 时间,无需额外排序,非常高效。
|
2023-02-06 19:57:19 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![二叉搜索树的中序遍历序列](binary_search_tree.assets/bst_inorder_traversal.png)
|
2023-02-06 19:57:19 +08:00
|
|
|
|
|
2023-02-16 03:39:01 +08:00
|
|
|
|
## 二叉搜索树的效率
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-04-17 18:22:18 +08:00
|
|
|
|
给定一组数据,我们考虑使用数组或二叉搜索树存储。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-04-17 18:22:18 +08:00
|
|
|
|
观察可知,二叉搜索树的各项操作的时间复杂度都是对数阶,具有稳定且高效的性能表现。只有在高频添加、低频查找删除的数据适用场景下,数组比二叉搜索树的效率更高。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
|
|
|
|
<div class="center-table" markdown>
|
|
|
|
|
|
2023-04-17 18:22:18 +08:00
|
|
|
|
| | 无序数组 | 二叉搜索树 |
|
|
|
|
|
| -------- | -------- | ----------- |
|
|
|
|
|
| 查找元素 | $O(n)$ | $O(\log n)$ |
|
|
|
|
|
| 插入元素 | $O(1)$ | $O(\log n)$ |
|
|
|
|
|
| 删除元素 | $O(n)$ | $O(\log n)$ |
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
|
|
|
|
</div>
|
|
|
|
|
|
2023-04-17 18:22:18 +08:00
|
|
|
|
在理想情况下,二叉搜索树是“平衡”的,这样就可以在 $\log n$ 轮循环内查找任意节点。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-04-10 23:59:22 +08:00
|
|
|
|
然而,如果我们在二叉搜索树中不断地插入和删除节点,可能导致二叉树退化为链表,这时各种操作的时间复杂度也会退化为 $O(n)$ 。
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-02-26 18:18:34 +08:00
|
|
|
|
![二叉搜索树的平衡与退化](binary_search_tree.assets/bst_degradation.png)
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-02-16 03:39:01 +08:00
|
|
|
|
## 二叉搜索树常见应用
|
2022-11-22 17:47:26 +08:00
|
|
|
|
|
2023-04-10 23:59:22 +08:00
|
|
|
|
- 用作系统中的多级索引,实现高效的查找、插入、删除操作。
|
|
|
|
|
- 作为某些搜索算法的底层数据结构。
|
|
|
|
|
- 用于存储数据流,以保持其有序状态。
|