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hello-algo/docs/chapter_tree/binary_tree_traversal.md

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Remove the heading numbers in all the source docs.
2023-02-16 03:39:01 +08:00
# 二叉树遍历
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
Polish the chapter of stack_and_queue, tree
2023-04-10 23:59:22 +08:00
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。
feat: add the section of Graph Traversal (#367) * Graph dev * Add the section of Graph Traversal. * Add missing Vertex.java * Add mkdocs.yml * Update numbering * Fix indentation and update array.md
2023-02-15 03:34:06 +08:00
Polish the chapter of stack_and_queue, tree
2023-04-10 23:59:22 +08:00
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。
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
Remove the heading numbers in all the source docs.
2023-02-16 03:39:01 +08:00
## 层序遍历
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
Mention figures and tables in normal texts. Fix some figures. Finetune texts.
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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。
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
层序遍历本质上属于「广度优先遍历 breadth-first traversal, BFS」，它体现了一种“一圈一圈向外扩展”的逐层遍历方式。
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
Update the captions of all the figures.
2023-02-26 18:18:34 +08:00
![二叉树的层序遍历](binary_tree_traversal.assets/binary_tree_bfs.png)
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
finetune
2023-08-29 20:57:37 +08:00
### 代码实现
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
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。实现代码如下：
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
Squash the language code blocks and fix list.md (#865)
2023-10-17 01:06:00 +08:00
```src
[file]{binary_tree_bfs}-[class]{}-[func]{level_order}
```
Add Zig code blocks.
2023-02-01 22:03:04 +08:00
finetune
2023-08-29 20:57:37 +08:00
### 复杂度分析
feat: add the section of Graph Traversal (#367) * Graph dev * Add the section of Graph Traversal. * Add missing Vertex.java * Add mkdocs.yml * Update numbering * Fix indentation and update array.md
2023-02-15 03:34:06 +08:00
finetune
2023-08-29 20:57:37 +08:00
- **时间复杂度 $O(n)$** ：所有节点被访问一次，使用 $O(n)$ 时间，其中 $n$ 为节点数量。
- **空间复杂度 $O(n)$** ：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。
feat: add the section of Graph Traversal (#367) * Graph dev * Add the section of Graph Traversal. * Add missing Vertex.java * Add mkdocs.yml * Update numbering * Fix indentation and update array.md
2023-02-15 03:34:06 +08:00
Remove the heading numbers in all the source docs.
2023-02-16 03:39:01 +08:00
## 前序、中序、后序遍历
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
相应地，前序、中序和后序遍历都属于「深度优先遍历 depth-first traversal, DFS」，它体现了一种“先走到尽头，再回溯继续”的遍历方式。
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
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棵二叉树的外围“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。
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
![二叉搜索树的前序、中序、后序遍历](binary_tree_traversal.assets/binary_tree_dfs.png)
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
finetune
2023-08-29 20:57:37 +08:00
### 代码实现
深度优先搜索通常基于递归实现：
Squash the language code blocks and fix list.md (#865)
2023-10-17 01:06:00 +08:00
```src
[file]{binary_tree_dfs}-[class]{}-[func]{post_order}
```
Add Zig code blocks.
2023-02-01 22:03:04 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
!!! tip
Update the structure of the chapter of binary tree.
2022-12-21 17:19:39 +08:00
feat: Revised the book (#978) * Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
2023-12-02 06:21:34 +08:00
深度优先搜索也可以基于迭代实现，有兴趣的读者可以自行研究。
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
Replace "：" with "。"
2023-08-27 22:49:47 +08:00
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。
Add the animation for the tree preorder traversal
2023-04-18 23:10:37 +08:00
Finetune the chapter of hashing, divide and conquer, backtracking, tree
2023-07-24 03:04:55 +08:00
1. “递”表示开启新方法，程序在此过程中访问下一个节点。
2. “归”表示函数返回，代表当前节点已经访问完毕。
Add the animation for the tree preorder traversal
2023-04-18 23:10:37 +08:00
=== "<1>"
Prepare 1.0.0b3 release.
2023-05-10 21:00:04 +08:00
![前序遍历的递归过程](binary_tree_traversal.assets/preorder_step1.png)
Add the animation for the tree preorder traversal
2023-04-18 23:10:37 +08:00
=== "<2>"
![preorder_step2](binary_tree_traversal.assets/preorder_step2.png)
=== "<3>"
![preorder_step3](binary_tree_traversal.assets/preorder_step3.png)
=== "<4>"
![preorder_step4](binary_tree_traversal.assets/preorder_step4.png)
=== "<5>"
![preorder_step5](binary_tree_traversal.assets/preorder_step5.png)
=== "<6>"
![preorder_step6](binary_tree_traversal.assets/preorder_step6.png)
=== "<7>"
![preorder_step7](binary_tree_traversal.assets/preorder_step7.png)
=== "<8>"
![preorder_step8](binary_tree_traversal.assets/preorder_step8.png)
=== "<9>"
![preorder_step9](binary_tree_traversal.assets/preorder_step9.png)
=== "<10>"
![preorder_step10](binary_tree_traversal.assets/preorder_step10.png)
=== "<11>"
![preorder_step11](binary_tree_traversal.assets/preorder_step11.png)
finetune
2023-08-29 20:57:37 +08:00
### 复杂度分析
- **时间复杂度 $O(n)$** ：所有节点被访问一次，使用 $O(n)$ 时间。
- **空间复杂度 $O(n)$** ：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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