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Revisit the English version (#1835)

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* Review the English version using Claude-4.5.

* Update mkdocs.yml

* Align the section titles.

* Bug fixes
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Yudong Jin
2025-12-30 17:54:01 +08:00
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en/docs/chapter_tree/binary_tree_traversal.md
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@@ -8,13 +8,13 @@ The common traversal methods for binary trees include level-order traversal, pre
As shown in the figure below, <u>level-order traversal</u> traverses the binary tree from top to bottom, layer by layer. Within each level, it visits nodes from left to right.
Level-order traversal is essentially a type of <u>breadth-first traversal</u>, also known as <u>breadth-first search (BFS)</u>, which embodies a "circumferentially outward expanding" layer-by-layer traversal method.
Level-order traversal is essentially <u>breadth-first traversal</u>, also known as <u>breadth-first search (BFS)</u>, which embodies a "expanding outward circle by circle" layer-by-layer traversal method.
![Level-order traversal of a binary tree](binary_tree_traversal.assets/binary_tree_bfs.png)
### Code implementation
Breadth-first traversal is usually implemented with the help of a "queue". The queue follows the "first in, first out" rule, while breadth-first traversal follows the "layer-by-layer progression" rule, the underlying ideas of the two are consistent. The implementation code is as follows:
Breadth-first traversal is typically implemented with the help of a "queue". The queue follows the "first in, first out" rule, while breadth-first traversal follows the "layer-by-layer progression" rule; the underlying ideas of the two are consistent. The implementation code is as follows:
```src
[file]{binary_tree_bfs}-[class]{}-[func]{level_order}
@@ -22,16 +22,16 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
### Complexity analysis
- **Time complexity is $O(n)$**: All nodes are visited once, taking $O(n)$ time, where $n$ is the number of nodes.
- **Space complexity is $O(n)$**: In the worst case, i.e., a full binary tree, before traversing to the bottom level, the queue can contain at most $(n + 1) / 2$ nodes simultaneously, occupying $O(n)$ space.
- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time, where $n$ is the number of nodes.
- **Space complexity is $O(n)$**: In the worst case, i.e., a full binary tree, before traversing to the bottom level, the queue contains at most $(n + 1) / 2$ nodes simultaneously, occupying $O(n)$ space.
## Preorder, in-order, and post-order traversal
## Preorder, inorder, and postorder traversal
Correspondingly, pre-order, in-order, and post-order traversal all belong to <u>depth-first traversal</u>, also known as <u>depth-first search (DFS)</u>, which embodies a "proceed to the end first, then backtrack and continue" traversal method.
Correspondingly, preorder, inorder, and postorder traversals all belong to <u>depth-first traversal</u>, also known as <u>depth-first search (DFS)</u>, which embodies a "first go to the end, then backtrack and continue" traversal method.
The figure below shows the working principle of performing a depth-first traversal on a binary tree. **Depth-first traversal is like "walking" around the entire binary tree**, encountering three positions at each node, corresponding to pre-order, in-order, and post-order traversal.
The figure below shows how depth-first traversal works on a binary tree. **Depth-first traversal is like "walking" around the perimeter of the entire binary tree**, encountering three positions at each node, corresponding to preorder, inorder, and postorder traversal.
![Preorder, in-order, and post-order traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png)
![Preorder, inorder, and postorder traversal of a binary tree](binary_tree_traversal.assets/binary_tree_dfs.png)
### Code implementation
@@ -45,13 +45,13 @@ Depth-first search is usually implemented based on recursion:
Depth-first search can also be implemented based on iteration, interested readers can study this on their own.
The figure below shows the recursive process of pre-order traversal of a binary tree, which can be divided into two opposite parts: "recursion" and "return".
The figure below shows the recursive process of preorder traversal of a binary tree, which can be divided into two opposite parts: "recursion" and "return".
1. "Recursion" means starting a new method, the program accesses the next node in this process.
2. "Return" means the function returns, indicating the current node has been fully accessed.
1. "Recursion" means opening a new method, where the program accesses the next node in this process.
2. "Return" means the function returns, indicating that the current node has been fully visited.
=== "<1>"
![The recursive process of pre-order traversal](binary_tree_traversal.assets/preorder_step1.png)
![The recursive process of preorder traversal](binary_tree_traversal.assets/preorder_step1.png)
=== "<2>"
![preorder_step2](binary_tree_traversal.assets/preorder_step2.png)
@@ -86,4 +86,4 @@ The figure below shows the recursive process of pre-order traversal of a binary
### Complexity analysis
- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time.
- **Space complexity is $O(n)$**: In the worst case, i.e., the tree degenerates into a linked list, the recursion depth reaches $n$, the system occupies $O(n)$ stack frame space.
- **Space complexity is $O(n)$**: In the worst case, i.e., the tree degenerates into a linked list, the recursion depth reaches $n$, and the system occupies $O(n)$ stack frame space.