--- comments: true --- # 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 breadth-first traversal, also known as breadth-first search (BFS), which proceeds outward level by level. ![Level-order traversal of a binary tree](binary_tree_traversal.assets/binary_tree_bfs.png){ class="animation-figure" }

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

### 1.   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: === "Python" ```python title="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 save the traversal sequence res = [] while queue: node: TreeNode = queue.popleft() # Dequeue res.append(node.val) # Save node value if node.left is not None: queue.append(node.left) # Left child node enqueue if node.right is not None: queue.append(node.right) # Right child node enqueue return res ``` === "C++" ```cpp title="binary_tree_bfs.cpp" /* Level-order traversal */ vector levelOrder(TreeNode *root) { // Initialize queue, add root node queue queue; queue.push(root); // Initialize a list to save the traversal sequence vector vec; while (!queue.empty()) { TreeNode *node = queue.front(); queue.pop(); // Dequeue vec.push_back(node->val); // Save node value if (node->left != nullptr) queue.push(node->left); // Left child node enqueue if (node->right != nullptr) queue.push(node->right); // Right child node enqueue } return vec; } ``` === "Java" ```java title="binary_tree_bfs.java" /* Level-order traversal */ List levelOrder(TreeNode root) { // Initialize queue, add root node Queue queue = new LinkedList<>(); queue.add(root); // Initialize a list to save the traversal sequence List list = new ArrayList<>(); while (!queue.isEmpty()) { TreeNode node = queue.poll(); // Dequeue list.add(node.val); // Save node value if (node.left != null) queue.offer(node.left); // Left child node enqueue if (node.right != null) queue.offer(node.right); // Right child node enqueue } return list; } ``` === "C#" ```csharp title="binary_tree_bfs.cs" /* Level-order traversal */ List LevelOrder(TreeNode root) { // Initialize queue, add root node Queue queue = new(); queue.Enqueue(root); // Initialize a list to save the traversal sequence List list = []; while (queue.Count != 0) { TreeNode node = queue.Dequeue(); // Dequeue list.Add(node.val!.Value); // Save node value if (node.left != null) queue.Enqueue(node.left); // Left child node enqueue if (node.right != null) queue.Enqueue(node.right); // Right child node enqueue } return list; } ``` === "Go" ```go title="binary_tree_bfs.go" /* Level-order traversal */ func levelOrder(root *TreeNode) []any { // Initialize queue, add root node queue := list.New() queue.PushBack(root) // Initialize a slice to save traversal sequence nums := make([]any, 0) for queue.Len() > 0 { // Dequeue node := queue.Remove(queue.Front()).(*TreeNode) // Save node value nums = append(nums, node.Val) if node.Left != nil { // Left child node enqueue queue.PushBack(node.Left) } if node.Right != nil { // Right child node enqueue queue.PushBack(node.Right) } } return nums } ``` === "Swift" ```swift title="binary_tree_bfs.swift" /* Level-order traversal */ func levelOrder(root: TreeNode) -> [Int] { // Initialize queue, add root node var queue: [TreeNode] = [root] // Initialize a list to save the traversal sequence var list: [Int] = [] while !queue.isEmpty { let node = queue.removeFirst() // Dequeue list.append(node.val) // Save node value if let left = node.left { queue.append(left) // Left child node enqueue } if let right = node.right { queue.append(right) // Right child node enqueue } } return list } ``` === "JS" ```javascript title="binary_tree_bfs.js" /* Level-order traversal */ function levelOrder(root) { // Initialize queue, add root node const queue = [root]; // Initialize a list to save the traversal sequence const list = []; while (queue.length) { let node = queue.shift(); // Dequeue list.push(node.val); // Save node value if (node.left) queue.push(node.left); // Left child node enqueue if (node.right) queue.push(node.right); // Right child node enqueue } return list; } ``` === "TS" ```typescript title="binary_tree_bfs.ts" /* Level-order traversal */ function levelOrder(root: TreeNode | null): number[] { // Initialize queue, add root node const queue = [root]; // Initialize a list to save the traversal sequence const list: number[] = []; while (queue.length) { let node = queue.shift() as TreeNode; // Dequeue list.push(node.val); // Save node value if (node.left) { queue.push(node.left); // Left child node enqueue } if (node.right) { queue.push(node.right); // Right child node enqueue } } return list; } ``` === "Dart" ```dart title="binary_tree_bfs.dart" /* Level-order traversal */ List levelOrder(TreeNode? root) { // Initialize queue, add root node Queue queue = Queue(); queue.add(root); // Initialize a list to save the traversal sequence List res = []; while (queue.isNotEmpty) { TreeNode? node = queue.removeFirst(); // Dequeue res.add(node!.val); // Save node value if (node.left != null) queue.add(node.left); // Left child node enqueue if (node.right != null) queue.add(node.right); // Right child node enqueue } return res; } ``` === "Rust" ```rust title="binary_tree_bfs.rs" /* Level-order traversal */ fn level_order(root: &Rc>) -> Vec { // Initialize queue, add root node let mut que = VecDeque::new(); que.push_back(root.clone()); // Initialize a list to save the traversal sequence let mut vec = Vec::new(); while let Some(node) = que.pop_front() { // Dequeue vec.push(node.borrow().val); // Save node value if let Some(left) = node.borrow().left.as_ref() { que.push_back(left.clone()); // Left child node enqueue } if let Some(right) = node.borrow().right.as_ref() { que.push_back(right.clone()); // Right child node enqueue }; } vec } ``` === "C" ```c title="binary_tree_bfs.c" /* Level-order traversal */ int *levelOrder(TreeNode *root, int *size) { /* Auxiliary queue */ int front, rear; int index, *arr; TreeNode *node; TreeNode **queue; /* Auxiliary queue */ queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE); // Queue pointer front = 0, rear = 0; // Add root node queue[rear++] = root; // Initialize a list to save the traversal sequence /* Auxiliary array */ arr = (int *)malloc(sizeof(int) * MAX_SIZE); // Array pointer index = 0; while (front < rear) { // Dequeue node = queue[front++]; // Save node value arr[index++] = node->val; if (node->left != NULL) { // Left child node enqueue queue[rear++] = node->left; } if (node->right != NULL) { // Right child node enqueue queue[rear++] = node->right; } } // Update array length value *size = index; arr = realloc(arr, sizeof(int) * (*size)); // Free auxiliary array space free(queue); return arr; } ``` === "Kotlin" ```kotlin title="binary_tree_bfs.kt" /* Level-order traversal */ fun levelOrder(root: TreeNode?): MutableList { // Initialize queue, add root node val queue = LinkedList() queue.add(root) // Initialize a list to save the traversal sequence val list = mutableListOf() while (queue.isNotEmpty()) { val node = queue.poll() // Dequeue list.add(node?._val!!) // Save node value if (node.left != null) queue.offer(node.left) // Left child node enqueue if (node.right != null) queue.offer(node.right) // Right child node enqueue } return list } ``` === "Ruby" ```ruby title="binary_tree_bfs.rb" ### Level-order traversal ### def level_order(root) # Initialize queue, add root node queue = [root] # Initialize a list to save the traversal sequence res = [] while !queue.empty? node = queue.shift # Dequeue res << node.val # Save node value queue << node.left unless node.left.nil? # Left child node enqueue queue << node.right unless node.right.nil? # Right child node enqueue end res end ``` ### 2.   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. ## 7.2.2   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 goes as deep as possible before backtracking. Figure 7-10 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. ![Preorder, inorder, and postorder traversal of a binary tree](binary_tree_traversal.assets/binary_tree_dfs.png){ class="animation-figure" }

Figure 7-10   Preorder, inorder, and postorder traversal of a binary tree

### 1.   Code Implementation Depth-first search is usually implemented based on recursion: === "Python" ```python title="binary_tree_dfs.py" def pre_order(root: TreeNode | None): """Preorder 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): """Inorder 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): """Postorder 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) ``` === "C++" ```cpp title="binary_tree_dfs.cpp" /* Preorder 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); } /* Inorder 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); } /* Postorder 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); } ``` === "Java" ```java title="binary_tree_dfs.java" /* Preorder 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); } /* Inorder 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); } /* Postorder 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); } ``` === "C#" ```csharp title="binary_tree_dfs.cs" /* Preorder traversal */ void PreOrder(TreeNode? root) { if (root == null) return; // Visit priority: root node -> left subtree -> right subtree list.Add(root.val!.Value); PreOrder(root.left); PreOrder(root.right); } /* Inorder traversal */ void InOrder(TreeNode? root) { if (root == null) return; // Visit priority: left subtree -> root node -> right subtree InOrder(root.left); list.Add(root.val!.Value); InOrder(root.right); } /* Postorder 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!.Value); } ``` === "Go" ```go title="binary_tree_dfs.go" /* Preorder traversal */ func preOrder(node *TreeNode) { if node == nil { return } // Visit priority: root node -> left subtree -> right subtree nums = append(nums, node.Val) preOrder(node.Left) preOrder(node.Right) } /* Inorder traversal */ func inOrder(node *TreeNode) { if node == nil { return } // Visit priority: left subtree -> root node -> right subtree inOrder(node.Left) nums = append(nums, node.Val) inOrder(node.Right) } /* Postorder traversal */ func postOrder(node *TreeNode) { if node == nil { return } // Visit priority: left subtree -> right subtree -> root node postOrder(node.Left) postOrder(node.Right) nums = append(nums, node.Val) } ``` === "Swift" ```swift title="binary_tree_dfs.swift" /* Preorder traversal */ func preOrder(root: TreeNode?) { guard let root = root else { return } // Visit priority: root node -> left subtree -> right subtree list.append(root.val) preOrder(root: root.left) preOrder(root: root.right) } /* Inorder traversal */ func inOrder(root: TreeNode?) { guard let root = root else { return } // Visit priority: left subtree -> root node -> right subtree inOrder(root: root.left) list.append(root.val) inOrder(root: root.right) } /* Postorder traversal */ func postOrder(root: TreeNode?) { guard let root = root else { return } // Visit priority: left subtree -> right subtree -> root node postOrder(root: root.left) postOrder(root: root.right) list.append(root.val) } ``` === "JS" ```javascript title="binary_tree_dfs.js" /* Preorder traversal */ function preOrder(root) { if (root === null) return; // Visit priority: root node -> left subtree -> right subtree list.push(root.val); preOrder(root.left); preOrder(root.right); } /* Inorder traversal */ function inOrder(root) { if (root === null) return; // Visit priority: left subtree -> root node -> right subtree inOrder(root.left); list.push(root.val); inOrder(root.right); } /* Postorder traversal */ function postOrder(root) { if (root === null) return; // Visit priority: left subtree -> right subtree -> root node postOrder(root.left); postOrder(root.right); list.push(root.val); } ``` === "TS" ```typescript title="binary_tree_dfs.ts" /* Preorder traversal */ function preOrder(root: TreeNode | null): void { if (root === null) { return; } // Visit priority: root node -> left subtree -> right subtree list.push(root.val); preOrder(root.left); preOrder(root.right); } /* Inorder traversal */ function inOrder(root: TreeNode | null): void { if (root === null) { return; } // Visit priority: left subtree -> root node -> right subtree inOrder(root.left); list.push(root.val); inOrder(root.right); } /* Postorder traversal */ function postOrder(root: TreeNode | null): void { if (root === null) { return; } // Visit priority: left subtree -> right subtree -> root node postOrder(root.left); postOrder(root.right); list.push(root.val); } ``` === "Dart" ```dart title="binary_tree_dfs.dart" /* Preorder traversal */ void preOrder(TreeNode? node) { if (node == null) return; // Visit priority: root node -> left subtree -> right subtree list.add(node.val); preOrder(node.left); preOrder(node.right); } /* Inorder traversal */ void inOrder(TreeNode? node) { if (node == null) return; // Visit priority: left subtree -> root node -> right subtree inOrder(node.left); list.add(node.val); inOrder(node.right); } /* Postorder traversal */ void postOrder(TreeNode? node) { if (node == null) return; // Visit priority: left subtree -> right subtree -> root node postOrder(node.left); postOrder(node.right); list.add(node.val); } ``` === "Rust" ```rust title="binary_tree_dfs.rs" /* Preorder traversal */ fn pre_order(root: Option<&Rc>>) -> Vec { let mut result = vec![]; fn dfs(root: Option<&Rc>>, res: &mut Vec) { if let Some(node) = root { // Visit priority: root node -> left subtree -> right subtree let node = node.borrow(); res.push(node.val); dfs(node.left.as_ref(), res); dfs(node.right.as_ref(), res); } } dfs(root, &mut result); result } /* Inorder traversal */ fn in_order(root: Option<&Rc>>) -> Vec { let mut result = vec![]; fn dfs(root: Option<&Rc>>, res: &mut Vec) { if let Some(node) = root { // Visit priority: left subtree -> root node -> right subtree let node = node.borrow(); dfs(node.left.as_ref(), res); res.push(node.val); dfs(node.right.as_ref(), res); } } dfs(root, &mut result); result } /* Postorder traversal */ fn post_order(root: Option<&Rc>>) -> Vec { let mut result = vec![]; fn dfs(root: Option<&Rc>>, res: &mut Vec) { if let Some(node) = root { // Visit priority: left subtree -> right subtree -> root node let node = node.borrow(); dfs(node.left.as_ref(), res); dfs(node.right.as_ref(), res); res.push(node.val); } } dfs(root, &mut result); result } ``` === "C" ```c title="binary_tree_dfs.c" /* Preorder traversal */ void preOrder(TreeNode *root, int *size) { if (root == NULL) return; // Visit priority: root node -> left subtree -> right subtree arr[(*size)++] = root->val; preOrder(root->left, size); preOrder(root->right, size); } /* Inorder traversal */ void inOrder(TreeNode *root, int *size) { if (root == NULL) return; // Visit priority: left subtree -> root node -> right subtree inOrder(root->left, size); arr[(*size)++] = root->val; inOrder(root->right, size); } /* Postorder traversal */ void postOrder(TreeNode *root, int *size) { if (root == NULL) return; // Visit priority: left subtree -> right subtree -> root node postOrder(root->left, size); postOrder(root->right, size); arr[(*size)++] = root->val; } ``` === "Kotlin" ```kotlin title="binary_tree_dfs.kt" /* Preorder traversal */ fun preOrder(root: TreeNode?) { if (root == null) return // Visit priority: root node -> left subtree -> right subtree list.add(root._val) preOrder(root.left) preOrder(root.right) } /* Inorder traversal */ fun inOrder(root: TreeNode?) { if (root == null) return // Visit priority: left subtree -> root node -> right subtree inOrder(root.left) list.add(root._val) inOrder(root.right) } /* Postorder traversal */ fun postOrder(root: TreeNode?) { if (root == null) return // Visit priority: left subtree -> right subtree -> root node postOrder(root.left) postOrder(root.right) list.add(root._val) } ``` === "Ruby" ```ruby title="binary_tree_dfs.rb" ### Pre-order traversal ### def pre_order(root) return if root.nil? # Visit priority: root node -> left subtree -> right subtree $res << root.val pre_order(root.left) pre_order(root.right) end ### In-order traversal ### def in_order(root) return if root.nil? # Visit priority: left subtree -> root node -> right subtree in_order(root.left) $res << root.val in_order(root.right) end ### Post-order traversal ### def post_order(root) return if root.nil? # Visit priority: left subtree -> right subtree -> root node post_order(root.left) post_order(root.right) $res << root.val end ``` !!! tip Depth-first search can also be implemented iteratively, and interested readers can explore this on their own. Figure 7-11 shows the recursive process of preorder traversal of a binary tree, which can be divided into two opposite phases: "descending" and "returning". 1. "Descending" means making a new recursive call, during which the program visits the next node. 2. "Returning" means the function call returns, indicating that the current node has been fully processed. === "<1>" ![The recursive process of preorder traversal](binary_tree_traversal.assets/preorder_step1.png){ class="animation-figure" } === "<2>" ![preorder_step2](binary_tree_traversal.assets/preorder_step2.png){ class="animation-figure" } === "<3>" ![preorder_step3](binary_tree_traversal.assets/preorder_step3.png){ class="animation-figure" } === "<4>" ![preorder_step4](binary_tree_traversal.assets/preorder_step4.png){ class="animation-figure" } === "<5>" ![preorder_step5](binary_tree_traversal.assets/preorder_step5.png){ class="animation-figure" } === "<6>" ![preorder_step6](binary_tree_traversal.assets/preorder_step6.png){ class="animation-figure" } === "<7>" ![preorder_step7](binary_tree_traversal.assets/preorder_step7.png){ class="animation-figure" } === "<8>" ![preorder_step8](binary_tree_traversal.assets/preorder_step8.png){ class="animation-figure" } === "<9>" ![preorder_step9](binary_tree_traversal.assets/preorder_step9.png){ class="animation-figure" } === "<10>" ![preorder_step10](binary_tree_traversal.assets/preorder_step10.png){ class="animation-figure" } === "<11>" ![preorder_step11](binary_tree_traversal.assets/preorder_step11.png){ class="animation-figure" }

Figure 7-11   The recursive process of preorder 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 $n$, and the system occupies $O(n)$ stack frame space.