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# 7.2 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.

## 7.2.1 Level-order traversal

As shown in Figure 7-9, 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.

Figure 7-9 Level-order traversal of a binary tree

### 1. 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:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby Zig
binary_tree_bfs.py

    def level_order(root: TreeNode | None) -> list[int]:
        """Level-order traversal"""
        # Initialize queue, add root node
        queue: deque[TreeNode] = deque()
        queue.append(root)
        # Initialize a list to store the traversal sequence
        res = []
        while queue:
            node: TreeNode = queue.popleft()  # Queue dequeues
            res.append(node.val)  # Save node value
            if node.left is not None:
                queue.append(node.left)  # Left child node enqueues
            if node.right is not None:
                queue.append(node.right)  # Right child node enqueues
        return res

binary_tree_bfs.cpp

    /* Level-order traversal */
    vector<int> levelOrder(TreeNode *root) {
        // Initialize queue, add root node
        queue<TreeNode *> queue;
        queue.push(root);
        // Initialize a list to store the traversal sequence
        vector<int> vec;
        while (!queue.empty()) {
            TreeNode *node = queue.front();
            queue.pop();              // Queue dequeues
            vec.push_back(node->val); // Save node value
            if (node->left != nullptr)
                queue.push(node->left); // Left child node enqueues
            if (node->right != nullptr)
                queue.push(node->right); // Right child node enqueues
        }
        return vec;
    }

binary_tree_bfs.java

    /* Level-order traversal */
    List<Integer> levelOrder(TreeNode root) {
        // Initialize queue, add root node
        Queue<TreeNode> queue = new LinkedList<>();
        queue.add(root);
        // Initialize a list to store the traversal sequence
        List<Integer> list = new ArrayList<>();
        while (!queue.isEmpty()) {
            TreeNode node = queue.poll(); // Queue dequeues
            list.add(node.val);           // Save node value
            if (node.left != null)
                queue.offer(node.left);   // Left child node enqueues
            if (node.right != null)
                queue.offer(node.right);  // Right child node enqueues
        }
        return list;
    }

binary_tree_bfs.cs

    [class]{binary_tree_bfs}-[func]{LevelOrder}

binary_tree_bfs.go

    [class]{}-[func]{levelOrder}

binary_tree_bfs.swift

    [class]{}-[func]{levelOrder}

binary_tree_bfs.js

    [class]{}-[func]{levelOrder}

binary_tree_bfs.ts

    [class]{}-[func]{levelOrder}

binary_tree_bfs.dart

    [class]{}-[func]{levelOrder}

binary_tree_bfs.rs

    [class]{}-[func]{level_order}

binary_tree_bfs.c

    [class]{}-[func]{levelOrder}

binary_tree_bfs.kt

    [class]{}-[func]{levelOrder}

binary_tree_bfs.rb

    [class]{}-[func]{level_order}

binary_tree_bfs.zig

    [class]{}-[func]{levelOrder}

### 2. 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.

## 7.2.2 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.

Figure 7-10 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.

Figure 7-10 Preorder, in-order, and post-order traversal of a binary search tree

### 1. Code implementation

Depth-first search is usually implemented based on recursion:

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby Zig
binary_tree_dfs.py

    def pre_order(root: TreeNode | None):
        """Pre-order traversal"""
        if root is None:
            return
        # Visit priority: root node -> left subtree -> right subtree
        res.append(root.val)
        pre_order(root=root.left)
        pre_order(root=root.right)

    def in_order(root: TreeNode | None):
        """In-order traversal"""
        if root is None:
            return
        # Visit priority: left subtree -> root node -> right subtree
        in_order(root=root.left)
        res.append(root.val)
        in_order(root=root.right)

    def post_order(root: TreeNode | None):
        """Post-order traversal"""
        if root is None:
            return
        # Visit priority: left subtree -> right subtree -> root node
        post_order(root=root.left)
        post_order(root=root.right)
        res.append(root.val)

binary_tree_dfs.cpp

    /* Pre-order traversal */
    void preOrder(TreeNode *root) {
        if (root == nullptr)
            return;
        // Visit priority: root node -> left subtree -> right subtree
        vec.push_back(root->val);
        preOrder(root->left);
        preOrder(root->right);
    }

    /* In-order traversal */
    void inOrder(TreeNode *root) {
        if (root == nullptr)
            return;
        // Visit priority: left subtree -> root node -> right subtree
        inOrder(root->left);
        vec.push_back(root->val);
        inOrder(root->right);
    }

    /* Post-order traversal */
    void postOrder(TreeNode *root) {
        if (root == nullptr)
            return;
        // Visit priority: left subtree -> right subtree -> root node
        postOrder(root->left);
        postOrder(root->right);
        vec.push_back(root->val);
    }

binary_tree_dfs.java

    /* Pre-order traversal */
    void preOrder(TreeNode root) {
        if (root == null)
            return;
        // Visit priority: root node -> left subtree -> right subtree
        list.add(root.val);
        preOrder(root.left);
        preOrder(root.right);
    }

    /* In-order traversal */
    void inOrder(TreeNode root) {
        if (root == null)
            return;
        // Visit priority: left subtree -> root node -> right subtree
        inOrder(root.left);
        list.add(root.val);
        inOrder(root.right);
    }

    /* Post-order traversal */
    void postOrder(TreeNode root) {
        if (root == null)
            return;
        // Visit priority: left subtree -> right subtree -> root node
        postOrder(root.left);
        postOrder(root.right);
        list.add(root.val);
    }

binary_tree_dfs.cs

    [class]{binary_tree_dfs}-[func]{PreOrder}

    [class]{binary_tree_dfs}-[func]{InOrder}

    [class]{binary_tree_dfs}-[func]{PostOrder}

binary_tree_dfs.go

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.swift

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.js

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.ts

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.dart

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.rs

    [class]{}-[func]{pre_order}

    [class]{}-[func]{in_order}

    [class]{}-[func]{post_order}

binary_tree_dfs.c

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.kt

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

binary_tree_dfs.rb

    [class]{}-[func]{pre_order}

    [class]{}-[func]{in_order}

    [class]{}-[func]{post_order}

binary_tree_dfs.zig

    [class]{}-[func]{preOrder}

    [class]{}-[func]{inOrder}

    [class]{}-[func]{postOrder}

Tip

Depth-first search can also be implemented based on iteration, interested readers can study this on their own.

Figure 7-11 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>

Figure 7-11 The recursive process of pre-order traversal

### 2. 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.

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