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

Binary tree traversal

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.

Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.

Level-order traversal

As shown in the figure below, level-order traversal traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.

Level-order traversal essentially belongs to 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, 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 lowest level, the queue can contain at most (n + 1) / 2 nodes at the same time, occupying O(n) space.

Preorder, inorder, and postorder traversal

Correspondingly, preorder, inorder, and postorder 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 perimeter of the entire binary tree, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder 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 preorder 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 degrades into a linked list, the recursion depth reaches n, the system occupies O(n) stack frame space.