hello-algo/en/docs/chapter_tree/binary_tree_traversal.md

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 a type of breadth-first traversal, also known as breadth-first search (BFS), 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:

[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 depth-first traversal, also known as depth-first search (DFS), 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:

[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.