mirror of
https://github.com/krahets/hello-algo.git
synced 2024-12-25 12:36:30 +08:00
89 lines
4.2 KiB
Markdown
Executable file
89 lines
4.2 KiB
Markdown
Executable file
# Binary tree traversal
|
|
|
|
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.
|
|
|
|
The common traversal methods for binary trees include level-order traversal, pre-order traversal, in-order traversal, and post-order traversal.
|
|
|
|
## Level-order traversal
|
|
|
|
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.
|
|
|
|
|
|
|
|
### 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:
|
|
|
|
```src
|
|
[file]{binary_tree_bfs}-[class]{}-[func]{level_order}
|
|
```
|
|
|
|
### 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.
|
|
|
|
## Preorder, in-order, and post-order 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.
|
|
|
|
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.
|
|
|
|
|
|
|
|
### Code implementation
|
|
|
|
Depth-first search is usually implemented based on recursion:
|
|
|
|
```src
|
|
[file]{binary_tree_dfs}-[class]{}-[func]{post_order}
|
|
```
|
|
|
|
!!! tip
|
|
|
|
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".
|
|
|
|
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>"
|
|
|
|
|
|
=== "<2>"
|
|
|
|
|
|
=== "<3>"
|
|
|
|
|
|
=== "<4>"
|
|
|
|
|
|
=== "<5>"
|
|
|
|
|
|
=== "<6>"
|
|
|
|
|
|
=== "<7>"
|
|
|
|
|
|
=== "<8>"
|
|
|
|
|
|
=== "<9>"
|
|
|
|
|
|
=== "<10>"
|
|
|
|
|
|
=== "<11>"
|
|
|
|
|
|
### 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.
|