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265 lines
6.3 KiB
Markdown
Executable file
265 lines
6.3 KiB
Markdown
Executable file
# 二叉树遍历
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从物理结构的角度来看，树是一种基于链表的数据结构，因此其遍历方式是通过指针逐个访问节点。然而，树是一种非线性数据结构，这使得遍历树比遍历链表更加复杂，需要借助搜索算法来实现。
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二叉树常见的遍历方式包括层序遍历、前序遍历、中序遍历和后序遍历等。
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## 层序遍历
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如下图所示，「层序遍历 level-order traversal」从顶部到底部逐层遍历二叉树，并在每一层按照从左到右的顺序访问节点。
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层序遍历本质上属于「广度优先遍历 breadth-first traversal」，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。
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广度优先遍历通常借助“队列”来实现。队列遵循“先进先出”的规则，而广度优先遍历则遵循“逐层推进”的规则，两者背后的思想是一致的。
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=== "Java"
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```java title="binary_tree_bfs.java"
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[class]{binary_tree_bfs}-[func]{levelOrder}
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```
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=== "C++"
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```cpp title="binary_tree_bfs.cpp"
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[class]{}-[func]{levelOrder}
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```
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=== "Python"
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```python title="binary_tree_bfs.py"
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[class]{}-[func]{level_order}
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```
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=== "Go"
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```go title="binary_tree_bfs.go"
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[class]{}-[func]{levelOrder}
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```
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=== "JS"
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```javascript title="binary_tree_bfs.js"
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[class]{}-[func]{levelOrder}
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```
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=== "TS"
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```typescript title="binary_tree_bfs.ts"
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[class]{}-[func]{levelOrder}
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```
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=== "C"
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```c title="binary_tree_bfs.c"
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[class]{}-[func]{levelOrder}
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```
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=== "C#"
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```csharp title="binary_tree_bfs.cs"
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[class]{binary_tree_bfs}-[func]{levelOrder}
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```
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=== "Swift"
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```swift title="binary_tree_bfs.swift"
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[class]{}-[func]{levelOrder}
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```
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=== "Zig"
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```zig title="binary_tree_bfs.zig"
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[class]{}-[func]{levelOrder}
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```
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=== "Dart"
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```dart title="binary_tree_bfs.dart"
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[class]{}-[func]{levelOrder}
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```
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=== "Rust"
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```rust title="binary_tree_bfs.rs"
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[class]{}-[func]{level_order}
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```
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**时间复杂度**：所有节点被访问一次，使用 $O(n)$ 时间，其中 $n$ 为节点数量。
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**空间复杂度**：在最差情况下，即满二叉树时，遍历到最底层之前，队列中最多同时存在 $(n + 1) / 2$ 个节点，占用 $O(n)$ 空间。
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## 前序、中序、后序遍历
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相应地，前序、中序和后序遍历都属于「深度优先遍历 depth-first traversal」，它体现了一种“先走到尽头，再回溯继续”的遍历方式。
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下图展示了对二叉树进行深度优先遍历的工作原理。**深度优先遍历就像是绕着整个二叉树的外围“走”一圈**，在每个节点都会遇到三个位置，分别对应前序遍历、中序遍历和后序遍历。
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=== "Java"
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```java title="binary_tree_dfs.java"
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[class]{binary_tree_dfs}-[func]{preOrder}
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[class]{binary_tree_dfs}-[func]{inOrder}
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[class]{binary_tree_dfs}-[func]{postOrder}
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```
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=== "C++"
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```cpp title="binary_tree_dfs.cpp"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Python"
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```python title="binary_tree_dfs.py"
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[class]{}-[func]{pre_order}
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[class]{}-[func]{in_order}
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[class]{}-[func]{post_order}
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```
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=== "Go"
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```go title="binary_tree_dfs.go"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "JS"
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```javascript title="binary_tree_dfs.js"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "TS"
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```typescript title="binary_tree_dfs.ts"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "C"
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```c title="binary_tree_dfs.c"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "C#"
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```csharp title="binary_tree_dfs.cs"
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[class]{binary_tree_dfs}-[func]{preOrder}
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[class]{binary_tree_dfs}-[func]{inOrder}
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[class]{binary_tree_dfs}-[func]{postOrder}
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```
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=== "Swift"
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```swift title="binary_tree_dfs.swift"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Zig"
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```zig title="binary_tree_dfs.zig"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Dart"
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```dart title="binary_tree_dfs.dart"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Rust"
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```rust title="binary_tree_dfs.rs"
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[class]{}-[func]{pre_order}
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[class]{}-[func]{in_order}
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[class]{}-[func]{post_order}
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```
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**时间复杂度**：所有节点被访问一次，使用 $O(n)$ 时间，其中 $n$ 为节点数量。
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**空间复杂度**：在最差情况下，即树退化为链表时，递归深度达到 $n$ ，系统占用 $O(n)$ 栈帧空间。
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!!! note
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我们也可以不使用递归，仅基于迭代实现前、中、后序遍历，有兴趣的同学可以自行实现。
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下图展示了前序遍历二叉树的递归过程，其可分为“递”和“归”两个逆向的部分：
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1. “递”表示开启新方法，程序在此过程中访问下一个节点。
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2. “归”表示函数返回，代表当前节点已经访问完毕。
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=== "<1>"
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=== "<2>"
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=== "<3>"
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=== "<4>"
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=== "<5>"
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=== "<6>"
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=== "<7>"
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=== "<8>"
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=== "<9>"
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=== "<10>"
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=== "<11>"
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