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* Add "reference" for EN version. Bug fixes. * Unify the figure reference as "the figure below" and "the figure above". Bug fixes. * Format the EN markdown files. * Replace "" with <u></u> for EN version and bug fixes * Fix biary_tree_dfs.png * Fix biary_tree_dfs.png * Fix zh-hant/biary_tree_dfs.png * Fix heap_sort_step1.png * Sync zh and zh-hant versions. * Bug fixes * Fix EN figures * Bug fixes * Fix the figure labels for EN version
89 lines
4.2 KiB
Markdown
Executable file
89 lines
4.2 KiB
Markdown
Executable file
# Binary tree traversal
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From the perspective of physical structure, a tree is a data structure based on linked lists, hence its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms to achieve.
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Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.
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## Level-order traversal
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As shown in the figure below, <u>level-order traversal</u> traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.
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Level-order traversal essentially belongs to <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.
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### Code implementation
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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:
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```src
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[file]{binary_tree_bfs}-[class]{}-[func]{level_order}
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```
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### Complexity analysis
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- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time, where $n$ is the number of nodes.
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- **Space complexity is $O(n)$**: In the worst case, i.e., a full binary tree, before traversing to the lowest level, the queue can contain at most $(n + 1) / 2$ nodes at the same time, occupying $O(n)$ space.
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## Preorder, inorder, and postorder traversal
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Correspondingly, preorder, inorder, and postorder 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.
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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 perimeter of the entire binary tree**, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder traversal.
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### Code implementation
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Depth-first search is usually implemented based on recursion:
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```src
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[file]{binary_tree_dfs}-[class]{}-[func]{post_order}
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```
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!!! tip
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Depth-first search can also be implemented based on iteration, interested readers can study this on their own.
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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".
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1. "Recursion" means starting a new method, the program accesses the next node in this process.
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2. "Return" means the function returns, indicating the current node has been fully accessed.
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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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### Complexity analysis
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- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time.
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- **Space complexity is $O(n)$**: In the worst case, i.e., the tree degrades into a linked list, the recursion depth reaches $n$, the system occupies $O(n)$ stack frame space.
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