--- comments: true --- # 7.2   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. ## 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, 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. ![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 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" ```python title="binary_tree_bfs.py" def level_order(root: TreeNode | None) -> list[int]: """层序遍历""" # 初始化队列,加入根节点 queue: deque[TreeNode] = deque() queue.append(root) # 初始化一个列表,用于保存遍历序列 res = [] while queue: node: TreeNode = queue.popleft() # 队列出队 res.append(node.val) # 保存节点值 if node.left is not None: queue.append(node.left) # 左子节点入队 if node.right is not None: queue.append(node.right) # 右子节点入队 return res ``` === "C++" ```cpp title="binary_tree_bfs.cpp" /* 层序遍历 */ vector levelOrder(TreeNode *root) { // 初始化队列,加入根节点 queue queue; queue.push(root); // 初始化一个列表,用于保存遍历序列 vector vec; while (!queue.empty()) { TreeNode *node = queue.front(); queue.pop(); // 队列出队 vec.push_back(node->val); // 保存节点值 if (node->left != nullptr) queue.push(node->left); // 左子节点入队 if (node->right != nullptr) queue.push(node->right); // 右子节点入队 } return vec; } ``` === "Java" ```java title="binary_tree_bfs.java" /* 层序遍历 */ List levelOrder(TreeNode root) { // 初始化队列,加入根节点 Queue queue = new LinkedList<>(); queue.add(root); // 初始化一个列表,用于保存遍历序列 List list = new ArrayList<>(); while (!queue.isEmpty()) { TreeNode node = queue.poll(); // 队列出队 list.add(node.val); // 保存节点值 if (node.left != null) queue.offer(node.left); // 左子节点入队 if (node.right != null) queue.offer(node.right); // 右子节点入队 } return list; } ``` === "C#" ```csharp title="binary_tree_bfs.cs" /* 层序遍历 */ List LevelOrder(TreeNode root) { // 初始化队列,加入根节点 Queue queue = new(); queue.Enqueue(root); // 初始化一个列表,用于保存遍历序列 List list = []; while (queue.Count != 0) { TreeNode node = queue.Dequeue(); // 队列出队 list.Add(node.val!.Value); // 保存节点值 if (node.left != null) queue.Enqueue(node.left); // 左子节点入队 if (node.right != null) queue.Enqueue(node.right); // 右子节点入队 } return list; } ``` === "Go" ```go title="binary_tree_bfs.go" /* 层序遍历 */ func levelOrder(root *TreeNode) []any { // 初始化队列,加入根节点 queue := list.New() queue.PushBack(root) // 初始化一个切片,用于保存遍历序列 nums := make([]any, 0) for queue.Len() > 0 { // 队列出队 node := queue.Remove(queue.Front()).(*TreeNode) // 保存节点值 nums = append(nums, node.Val) if node.Left != nil { // 左子节点入队 queue.PushBack(node.Left) } if node.Right != nil { // 右子节点入队 queue.PushBack(node.Right) } } return nums } ``` === "Swift" ```swift title="binary_tree_bfs.swift" /* 层序遍历 */ func levelOrder(root: TreeNode) -> [Int] { // 初始化队列,加入根节点 var queue: [TreeNode] = [root] // 初始化一个列表,用于保存遍历序列 var list: [Int] = [] while !queue.isEmpty { let node = queue.removeFirst() // 队列出队 list.append(node.val) // 保存节点值 if let left = node.left { queue.append(left) // 左子节点入队 } if let right = node.right { queue.append(right) // 右子节点入队 } } return list } ``` === "JS" ```javascript title="binary_tree_bfs.js" /* 层序遍历 */ function levelOrder(root) { // 初始化队列,加入根节点 const queue = [root]; // 初始化一个列表,用于保存遍历序列 const list = []; while (queue.length) { let node = queue.shift(); // 队列出队 list.push(node.val); // 保存节点值 if (node.left) queue.push(node.left); // 左子节点入队 if (node.right) queue.push(node.right); // 右子节点入队 } return list; } ``` === "TS" ```typescript title="binary_tree_bfs.ts" /* 层序遍历 */ function levelOrder(root: TreeNode | null): number[] { // 初始化队列,加入根节点 const queue = [root]; // 初始化一个列表,用于保存遍历序列 const list: number[] = []; while (queue.length) { let node = queue.shift() as TreeNode; // 队列出队 list.push(node.val); // 保存节点值 if (node.left) { queue.push(node.left); // 左子节点入队 } if (node.right) { queue.push(node.right); // 右子节点入队 } } return list; } ``` === "Dart" ```dart title="binary_tree_bfs.dart" /* 层序遍历 */ List levelOrder(TreeNode? root) { // 初始化队列,加入根节点 Queue queue = Queue(); queue.add(root); // 初始化一个列表,用于保存遍历序列 List res = []; while (queue.isNotEmpty) { TreeNode? node = queue.removeFirst(); // 队列出队 res.add(node!.val); // 保存节点值 if (node.left != null) queue.add(node.left); // 左子节点入队 if (node.right != null) queue.add(node.right); // 右子节点入队 } return res; } ``` === "Rust" ```rust title="binary_tree_bfs.rs" /* 层序遍历 */ fn level_order(root: &Rc>) -> Vec { // 初始化队列,加入根节点 let mut que = VecDeque::new(); que.push_back(root.clone()); // 初始化一个列表,用于保存遍历序列 let mut vec = Vec::new(); while let Some(node) = que.pop_front() { // 队列出队 vec.push(node.borrow().val); // 保存节点值 if let Some(left) = node.borrow().left.as_ref() { que.push_back(left.clone()); // 左子节点入队 } if let Some(right) = node.borrow().right.as_ref() { que.push_back(right.clone()); // 右子节点入队 }; } vec } ``` === "C" ```c title="binary_tree_bfs.c" /* 层序遍历 */ int *levelOrder(TreeNode *root, int *size) { /* 辅助队列 */ int front, rear; int index, *arr; TreeNode *node; TreeNode **queue; /* 辅助队列 */ queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE); // 队列指针 front = 0, rear = 0; // 加入根节点 queue[rear++] = root; // 初始化一个列表,用于保存遍历序列 /* 辅助数组 */ arr = (int *)malloc(sizeof(int) * MAX_SIZE); // 数组指针 index = 0; while (front < rear) { // 队列出队 node = queue[front++]; // 保存节点值 arr[index++] = node->val; if (node->left != NULL) { // 左子节点入队 queue[rear++] = node->left; } if (node->right != NULL) { // 右子节点入队 queue[rear++] = node->right; } } // 更新数组长度的值 *size = index; arr = realloc(arr, sizeof(int) * (*size)); // 释放辅助数组空间 free(queue); return arr; } ``` === "Kotlin" ```kotlin title="binary_tree_bfs.kt" /* 层序遍历 */ fun levelOrder(root: TreeNode?): MutableList { // 初始化队列,加入根节点 val queue = LinkedList() queue.add(root) // 初始化一个列表,用于保存遍历序列 val list = mutableListOf() while (queue.isNotEmpty()) { val node = queue.poll() // 队列出队 list.add(node?._val!!) // 保存节点值 if (node.left != null) queue.offer(node.left) // 左子节点入队 if (node.right != null) queue.offer(node.right) // 右子节点入队 } return list } ``` === "Ruby" ```ruby title="binary_tree_bfs.rb" ### 层序遍历 ### def level_order(root) # 初始化队列,加入根节点 queue = [root] # 初始化一个列表,用于保存遍历序列 res = [] while !queue.empty? node = queue.shift # 队列出队 res << node.val # 保存节点值 queue << node.left unless node.left.nil? # 左子节点入队 queue << node.right unless node.right.nil? # 右子节点入队 end res end ``` === "Zig" ```zig title="binary_tree_bfs.zig" // 层序遍历 fn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) { // 初始化队列,加入根节点 const L = std.TailQueue(*inc.TreeNode(T)); var queue = L{}; var root_node = try mem_allocator.create(L.Node); root_node.data = root; queue.append(root_node); // 初始化一个列表,用于保存遍历序列 var list = std.ArrayList(T).init(std.heap.page_allocator); while (queue.len > 0) { var queue_node = queue.popFirst().?; // 队列出队 var node = queue_node.data; try list.append(node.val); // 保存节点值 if (node.left != null) { var tmp_node = try mem_allocator.create(L.Node); tmp_node.data = node.left.?; queue.append(tmp_node); // 左子节点入队 } if (node.right != null) { var tmp_node = try mem_allocator.create(L.Node); tmp_node.data = node.right.?; queue.append(tmp_node); // 右子节点入队 } } return list; } ``` ??? pythontutor "Code Visualization"
### 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 lowest level, the queue can contain at most $(n + 1) / 2$ nodes at the same time, occupying $O(n)$ space. ## 7.2.2   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. 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 perimeter of the entire binary tree**, encountering three positions at each node, corresponding to preorder traversal, inorder traversal, and postorder traversal. ![Preorder, inorder, and postorder traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png){ class="animation-figure" }

Figure 7-10   Preorder, inorder, and postorder traversal of a binary search 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): """前序遍历""" if root is None: return # 访问优先级:根节点 -> 左子树 -> 右子树 res.append(root.val) pre_order(root=root.left) pre_order(root=root.right) def in_order(root: TreeNode | None): """中序遍历""" if root is None: return # 访问优先级:左子树 -> 根节点 -> 右子树 in_order(root=root.left) res.append(root.val) in_order(root=root.right) def post_order(root: TreeNode | None): """后序遍历""" if root is None: return # 访问优先级:左子树 -> 右子树 -> 根节点 post_order(root=root.left) post_order(root=root.right) res.append(root.val) ``` === "C++" ```cpp title="binary_tree_dfs.cpp" /* 前序遍历 */ void preOrder(TreeNode *root) { if (root == nullptr) return; // 访问优先级:根节点 -> 左子树 -> 右子树 vec.push_back(root->val); preOrder(root->left); preOrder(root->right); } /* 中序遍历 */ void inOrder(TreeNode *root) { if (root == nullptr) return; // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root->left); vec.push_back(root->val); inOrder(root->right); } /* 后序遍历 */ void postOrder(TreeNode *root) { if (root == nullptr) return; // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root->left); postOrder(root->right); vec.push_back(root->val); } ``` === "Java" ```java title="binary_tree_dfs.java" /* 前序遍历 */ void preOrder(TreeNode root) { if (root == null) return; // 访问优先级:根节点 -> 左子树 -> 右子树 list.add(root.val); preOrder(root.left); preOrder(root.right); } /* 中序遍历 */ void inOrder(TreeNode root) { if (root == null) return; // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root.left); list.add(root.val); inOrder(root.right); } /* 后序遍历 */ void postOrder(TreeNode root) { if (root == null) return; // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root.left); postOrder(root.right); list.add(root.val); } ``` === "C#" ```csharp title="binary_tree_dfs.cs" /* 前序遍历 */ void PreOrder(TreeNode? root) { if (root == null) return; // 访问优先级:根节点 -> 左子树 -> 右子树 list.Add(root.val!.Value); PreOrder(root.left); PreOrder(root.right); } /* 中序遍历 */ void InOrder(TreeNode? root) { if (root == null) return; // 访问优先级:左子树 -> 根节点 -> 右子树 InOrder(root.left); list.Add(root.val!.Value); InOrder(root.right); } /* 后序遍历 */ void PostOrder(TreeNode? root) { if (root == null) return; // 访问优先级:左子树 -> 右子树 -> 根节点 PostOrder(root.left); PostOrder(root.right); list.Add(root.val!.Value); } ``` === "Go" ```go title="binary_tree_dfs.go" /* 前序遍历 */ func preOrder(node *TreeNode) { if node == nil { return } // 访问优先级:根节点 -> 左子树 -> 右子树 nums = append(nums, node.Val) preOrder(node.Left) preOrder(node.Right) } /* 中序遍历 */ func inOrder(node *TreeNode) { if node == nil { return } // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(node.Left) nums = append(nums, node.Val) inOrder(node.Right) } /* 后序遍历 */ func postOrder(node *TreeNode) { if node == nil { return } // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(node.Left) postOrder(node.Right) nums = append(nums, node.Val) } ``` === "Swift" ```swift title="binary_tree_dfs.swift" /* 前序遍历 */ func preOrder(root: TreeNode?) { guard let root = root else { return } // 访问优先级:根节点 -> 左子树 -> 右子树 list.append(root.val) preOrder(root: root.left) preOrder(root: root.right) } /* 中序遍历 */ func inOrder(root: TreeNode?) { guard let root = root else { return } // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root: root.left) list.append(root.val) inOrder(root: root.right) } /* 后序遍历 */ func postOrder(root: TreeNode?) { guard let root = root else { return } // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root: root.left) postOrder(root: root.right) list.append(root.val) } ``` === "JS" ```javascript title="binary_tree_dfs.js" /* 前序遍历 */ function preOrder(root) { if (root === null) return; // 访问优先级:根节点 -> 左子树 -> 右子树 list.push(root.val); preOrder(root.left); preOrder(root.right); } /* 中序遍历 */ function inOrder(root) { if (root === null) return; // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root.left); list.push(root.val); inOrder(root.right); } /* 后序遍历 */ function postOrder(root) { if (root === null) return; // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root.left); postOrder(root.right); list.push(root.val); } ``` === "TS" ```typescript title="binary_tree_dfs.ts" /* 前序遍历 */ function preOrder(root: TreeNode | null): void { if (root === null) { return; } // 访问优先级:根节点 -> 左子树 -> 右子树 list.push(root.val); preOrder(root.left); preOrder(root.right); } /* 中序遍历 */ function inOrder(root: TreeNode | null): void { if (root === null) { return; } // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root.left); list.push(root.val); inOrder(root.right); } /* 后序遍历 */ function postOrder(root: TreeNode | null): void { if (root === null) { return; } // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root.left); postOrder(root.right); list.push(root.val); } ``` === "Dart" ```dart title="binary_tree_dfs.dart" /* 前序遍历 */ void preOrder(TreeNode? node) { if (node == null) return; // 访问优先级:根节点 -> 左子树 -> 右子树 list.add(node.val); preOrder(node.left); preOrder(node.right); } /* 中序遍历 */ void inOrder(TreeNode? node) { if (node == null) return; // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(node.left); list.add(node.val); inOrder(node.right); } /* 后序遍历 */ void postOrder(TreeNode? node) { if (node == null) return; // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(node.left); postOrder(node.right); list.add(node.val); } ``` === "Rust" ```rust title="binary_tree_dfs.rs" /* 前序遍历 */ fn pre_order(root: Option<&Rc>>) -> Vec { let mut result = vec![]; if let Some(node) = root { // 访问优先级:根节点 -> 左子树 -> 右子树 result.push(node.borrow().val); result.extend(pre_order(node.borrow().left.as_ref())); result.extend(pre_order(node.borrow().right.as_ref())); } result } /* 中序遍历 */ fn in_order(root: Option<&Rc>>) -> Vec { let mut result = vec![]; if let Some(node) = root { // 访问优先级:左子树 -> 根节点 -> 右子树 result.extend(in_order(node.borrow().left.as_ref())); result.push(node.borrow().val); result.extend(in_order(node.borrow().right.as_ref())); } result } /* 后序遍历 */ fn post_order(root: Option<&Rc>>) -> Vec { let mut result = vec![]; if let Some(node) = root { // 访问优先级:左子树 -> 右子树 -> 根节点 result.extend(post_order(node.borrow().left.as_ref())); result.extend(post_order(node.borrow().right.as_ref())); result.push(node.borrow().val); } result } ``` === "C" ```c title="binary_tree_dfs.c" /* 前序遍历 */ void preOrder(TreeNode *root, int *size) { if (root == NULL) return; // 访问优先级:根节点 -> 左子树 -> 右子树 arr[(*size)++] = root->val; preOrder(root->left, size); preOrder(root->right, size); } /* 中序遍历 */ void inOrder(TreeNode *root, int *size) { if (root == NULL) return; // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root->left, size); arr[(*size)++] = root->val; inOrder(root->right, size); } /* 后序遍历 */ void postOrder(TreeNode *root, int *size) { if (root == NULL) return; // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root->left, size); postOrder(root->right, size); arr[(*size)++] = root->val; } ``` === "Kotlin" ```kotlin title="binary_tree_dfs.kt" /* 前序遍历 */ fun preOrder(root: TreeNode?) { if (root == null) return // 访问优先级:根节点 -> 左子树 -> 右子树 list.add(root._val) preOrder(root.left) preOrder(root.right) } /* 中序遍历 */ fun inOrder(root: TreeNode?) { if (root == null) return // 访问优先级:左子树 -> 根节点 -> 右子树 inOrder(root.left) list.add(root._val) inOrder(root.right) } /* 后序遍历 */ fun postOrder(root: TreeNode?) { if (root == null) return // 访问优先级:左子树 -> 右子树 -> 根节点 postOrder(root.left) postOrder(root.right) list.add(root._val) } ``` === "Ruby" ```ruby title="binary_tree_dfs.rb" ### 前序遍历 ### def pre_order(root) return if root.nil? # 访问优先级:根节点 -> 左子树 -> 右子树 $res << root.val pre_order(root.left) pre_order(root.right) end ### 中序遍历 ### def in_order(root) return if root.nil? # 访问优先级:左子树 -> 根节点 -> 右子树 in_order(root.left) $res << root.val in_order(root.right) end ### 后序遍历 ### def post_order(root) return if root.nil? # 访问优先级:左子树 -> 右子树 -> 根节点 post_order(root.left) post_order(root.right) $res << root.val end ``` === "Zig" ```zig title="binary_tree_dfs.zig" // 前序遍历 fn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void { if (root == null) return; // 访问优先级:根节点 -> 左子树 -> 右子树 try list.append(root.?.val); try preOrder(T, root.?.left); try preOrder(T, root.?.right); } // 中序遍历 fn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void { if (root == null) return; // 访问优先级:左子树 -> 根节点 -> 右子树 try inOrder(T, root.?.left); try list.append(root.?.val); try inOrder(T, root.?.right); } // 后序遍历 fn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void { if (root == null) return; // 访问优先级:左子树 -> 右子树 -> 根节点 try postOrder(T, root.?.left); try postOrder(T, root.?.right); try list.append(root.?.val); } ``` ??? pythontutor "Code Visualization"
!!! 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 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>" ![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 degrades into a linked list, the recursion depth reaches $n$, the system occupies $O(n)$ stack frame space.