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89 lines
4.2 KiB
Markdown
Executable file
89 lines
4.2 KiB
Markdown
Executable file
# Binary tree traversal
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From a physical structure perspective, 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.
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The common traversal methods for binary trees include level-order traversal, pre-order traversal, in-order traversal, and post-order traversal.
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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. Within each level, it visits nodes from left to right.
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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.
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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, taking $O(n)$ 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 bottom level, the queue can contain at most $(n + 1) / 2$ nodes simultaneously, occupying $O(n)$ space.
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## Preorder, in-order, and post-order traversal
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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.
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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 entire binary tree**, encountering three positions at each node, corresponding to pre-order, in-order, and post-order 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 pre-order 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 degenerates into a linked list, the recursion depth reaches $n$, the system occupies $O(n)$ stack frame space.
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