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, level-order traversal 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 breadth-first traversal, also known as breadth-first search (BFS), which embodies a "expanding outward circle by circle" layer-by-layer traversal method.

Code Implementation

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}

Complexity Analysis

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, Inorder, and Postorder Traversal

Correspondingly, preorder, inorder, and postorder traversals all belong to depth-first traversal, also known as depth-first search (DFS), which embodies a "first go to the end, then backtrack and continue" traversal method.

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.

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 preorder traversal of a binary tree, which can be divided into two opposite parts: "recursion" and "return".

"Recursion" means opening a new method, where the program accesses the next node in this process.
"Return" means the function returns, indicating that the current node has been fully visited.

=== "<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 $n$, and the system occupies $O(n)$ stack frame space.