7.2 Traversing a binary tree¶
In its physical representation, a tree is a linked data structure, so traversal follows pointers from node to node. Unlike a linked list, however, a tree is nonlinear. Traversing it is more complex and requires a search algorithm.
Common binary-tree traversals include level-order, preorder, inorder, and postorder traversal.
7.2.1 Level-order traversal¶
As shown below, level-order traversal visits the tree level by level from top to bottom, moving from left to right within each level.
Level-order traversal is a form of breadth-first traversal, also called breadth-first search (BFS). It explores outward one layer at a time.
Figure 7-9 Level-order traversal of a binary tree
1. Implementation¶
Breadth-first traversal usually uses a queue. A queue's first-in, first-out rule matches the traversal's level-by-level progression. The implementation follows:
binary_tree_bfs.py
def level_order(root: TreeNode | None) -> list[int]:
"""Level-order traversal"""
# Initialize a queue and enqueue the root
queue: deque[TreeNode] = deque()
queue.append(root)
# Initialize a list to store the traversal sequence
res = []
while queue:
node: TreeNode = queue.popleft() # Dequeue from the queue
res.append(node.val) # Store the node value
if node.left is not None:
queue.append(node.left) # Enqueue the left child
if node.right is not None:
queue.append(node.right) # Enqueue the right child
return res
binary_tree_bfs.cpp
/* Level-order traversal */
vector<int> levelOrder(TreeNode *root) {
// Initialize a queue and enqueue the root
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(); // Dequeue from the queue
vec.push_back(node->val); // Store the node value
if (node->left != nullptr)
queue.push(node->left); // Enqueue the left child
if (node->right != nullptr)
queue.push(node->right); // Enqueue the right child
}
return vec;
}
binary_tree_bfs.java
/* Level-order traversal */
List<Integer> levelOrder(TreeNode root) {
// Initialize a queue and enqueue the root
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(); // Dequeue from the queue
list.add(node.val); // Store the node value
if (node.left != null)
queue.offer(node.left); // Enqueue the left child
if (node.right != null)
queue.offer(node.right); // Enqueue the right child
}
return list;
}
binary_tree_bfs.cs
/* Level-order traversal */
List<int> LevelOrder(TreeNode root) {
// Initialize a queue and enqueue the root
Queue<TreeNode> queue = new();
queue.Enqueue(root);
// Initialize a list to store the traversal sequence
List<int> list = [];
while (queue.Count != 0) {
TreeNode node = queue.Dequeue(); // Dequeue from the queue
list.Add(node.val!.Value); // Store the node value
if (node.left != null)
queue.Enqueue(node.left); // Enqueue the left child
if (node.right != null)
queue.Enqueue(node.right); // Enqueue the right child
}
return list;
}
binary_tree_bfs.go
/* Level-order traversal */
func levelOrder(root *TreeNode) []any {
// Initialize a queue and enqueue the root
queue := list.New()
queue.PushBack(root)
// Initialize a slice to store the traversal sequence
nums := make([]any, 0)
for queue.Len() > 0 {
// Dequeue from the queue
node := queue.Remove(queue.Front()).(*TreeNode)
// Store the node value
nums = append(nums, node.Val)
if node.Left != nil {
// Enqueue the left child
queue.PushBack(node.Left)
}
if node.Right != nil {
// Enqueue the right child
queue.PushBack(node.Right)
}
}
return nums
}
binary_tree_bfs.swift
/* Level-order traversal */
func levelOrder(root: TreeNode) -> [Int] {
// Initialize a queue and enqueue the root
var queue: [TreeNode] = [root]
// Initialize a list to store the traversal sequence
var list: [Int] = []
while !queue.isEmpty {
let node = queue.removeFirst() // Dequeue from the queue
list.append(node.val) // Store the node value
if let left = node.left {
queue.append(left) // Enqueue the left child
}
if let right = node.right {
queue.append(right) // Enqueue the right child
}
}
return list
}
binary_tree_bfs.js
/* Level-order traversal */
function levelOrder(root) {
// Initialize a queue and enqueue the root
const queue = [root];
// Initialize a list to store the traversal sequence
const list = [];
while (queue.length) {
let node = queue.shift(); // Dequeue from the queue
list.push(node.val); // Store the node value
if (node.left) queue.push(node.left); // Enqueue the left child
if (node.right) queue.push(node.right); // Enqueue the right child
}
return list;
}
binary_tree_bfs.ts
/* Level-order traversal */
function levelOrder(root: TreeNode | null): number[] {
// Initialize a queue and enqueue the root
const queue = [root];
// Initialize a list to store the traversal sequence
const list: number[] = [];
while (queue.length) {
let node = queue.shift() as TreeNode; // Dequeue from the queue
list.push(node.val); // Store the node value
if (node.left) {
queue.push(node.left); // Enqueue the left child
}
if (node.right) {
queue.push(node.right); // Enqueue the right child
}
}
return list;
}
binary_tree_bfs.dart
/* Level-order traversal */
List<int> levelOrder(TreeNode? root) {
// Initialize a queue and enqueue the root
Queue<TreeNode?> queue = Queue();
queue.add(root);
// Initialize a list to store the traversal sequence
List<int> res = [];
while (queue.isNotEmpty) {
TreeNode? node = queue.removeFirst(); // Dequeue from the queue
res.add(node!.val); // Store the node value
if (node.left != null) queue.add(node.left); // Enqueue the left child
if (node.right != null) queue.add(node.right); // Enqueue the right child
}
return res;
}
binary_tree_bfs.rs
/* Level-order traversal */
fn level_order(root: &Rc<RefCell<TreeNode>>) -> Vec<i32> {
// Initialize a queue and enqueue the root
let mut que = VecDeque::new();
que.push_back(root.clone());
// Initialize a list to store the traversal sequence
let mut vec = Vec::new();
while let Some(node) = que.pop_front() {
// Dequeue from the queue
vec.push(node.borrow().val); // Store the node value
if let Some(left) = node.borrow().left.as_ref() {
que.push_back(left.clone()); // Enqueue the left child
}
if let Some(right) = node.borrow().right.as_ref() {
que.push_back(right.clone()); // Enqueue the right child
};
}
vec
}
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 indices
front = 0, rear = 0;
// Enqueue the root node
queue[rear++] = root;
// Initialize a list to store the traversal sequence
/* Auxiliary array */
arr = (int *)malloc(sizeof(int) * MAX_SIZE);
// Array index
index = 0;
while (front < rear) {
// Dequeue from the queue
node = queue[front++];
// Store the node value
arr[index++] = node->val;
if (node->left != NULL) {
// Enqueue the left child
queue[rear++] = node->left;
}
if (node->right != NULL) {
// Enqueue the right child
queue[rear++] = node->right;
}
}
// Update the array length
*size = index;
arr = realloc(arr, sizeof(int) * (*size));
// Free the auxiliary queue array
free(queue);
return arr;
}
binary_tree_bfs.kt
/* Level-order traversal */
fun levelOrder(root: TreeNode?): MutableList<Int> {
// Initialize a queue and enqueue the root
val queue = LinkedList<TreeNode?>()
queue.add(root)
// Initialize a list to store the traversal sequence
val list = mutableListOf<Int>()
while (queue.isNotEmpty()) {
val node = queue.poll() // Dequeue from the queue
list.add(node?._val!!) // Store the node value
if (node.left != null)
queue.offer(node.left) // Enqueue the left child
if (node.right != null)
queue.offer(node.right) // Enqueue the right child
}
return list
}
binary_tree_bfs.rb
### Level-order traversal ###
def level_order(root)
# Initialize a queue and enqueue the root
queue = [root]
# Initialize a list to store the traversal sequence
res = []
while !queue.empty?
node = queue.shift # Dequeue from the queue
res << node.val # Store the node value
queue << node.left unless node.left.nil? # Enqueue the left child
queue << node.right unless node.right.nil? # Enqueue the right child
end
res
end
Visualize execution
2. Complexity analysis¶
- Time complexity: \(O(n)\). Each of the \(n\) nodes is visited once, taking \(O(n)\) time.
- Space complexity: \(O(n)\). In the worst case, a perfect binary tree, the queue holds up to \((n + 1) / 2\) nodes just before the lowest level is visited, requiring \(O(n)\) space.
7.2.2 Preorder, inorder, and postorder traversal¶
Preorder, inorder, and postorder traversal are forms of depth-first traversal, also called depth-first search (DFS). They follow a path to its end, then backtrack and continue.
Figure 7-10 illustrates depth-first traversal. Imagine walking once around the outside of the entire binary tree. At each node, you encounter three positions corresponding to preorder, inorder, and postorder visits.
Figure 7-10 Preorder, inorder, and postorder traversal of a binary search tree
1. Implementation¶
Depth-first search is usually implemented recursively:
binary_tree_dfs.py
def pre_order(root: TreeNode | None):
"""Preorder traversal"""
if root is None:
return
# Visit order: root -> 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 order: left subtree -> root -> 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 order: left subtree -> right subtree -> root
post_order(root=root.left)
post_order(root=root.right)
res.append(root.val)
binary_tree_dfs.cpp
/* Preorder traversal */
void preOrder(TreeNode *root) {
if (root == nullptr)
return;
// Visit order: root -> 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 order: left subtree -> root -> right subtree
inOrder(root->left);
vec.push_back(root->val);
inOrder(root->right);
}
/* Postorder traversal */
void postOrder(TreeNode *root) {
if (root == nullptr)
return;
// Visit order: left subtree -> right subtree -> root
postOrder(root->left);
postOrder(root->right);
vec.push_back(root->val);
}
binary_tree_dfs.java
/* Preorder traversal */
void preOrder(TreeNode root) {
if (root == null)
return;
// Visit order: root -> 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 order: left subtree -> root -> right subtree
inOrder(root.left);
list.add(root.val);
inOrder(root.right);
}
/* Postorder traversal */
void postOrder(TreeNode root) {
if (root == null)
return;
// Visit order: left subtree -> right subtree -> root
postOrder(root.left);
postOrder(root.right);
list.add(root.val);
}
binary_tree_dfs.cs
/* Preorder traversal */
void PreOrder(TreeNode? root) {
if (root == null) return;
// Visit order: root -> 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 order: left subtree -> root -> right subtree
InOrder(root.left);
list.Add(root.val!.Value);
InOrder(root.right);
}
/* Postorder traversal */
void PostOrder(TreeNode? root) {
if (root == null) return;
// Visit order: left subtree -> right subtree -> root
PostOrder(root.left);
PostOrder(root.right);
list.Add(root.val!.Value);
}
binary_tree_dfs.go
/* Preorder traversal */
func preOrder(node *TreeNode) {
if node == nil {
return
}
// Visit order: root -> 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 order: left subtree -> root -> right subtree
inOrder(node.Left)
nums = append(nums, node.Val)
inOrder(node.Right)
}
/* Postorder traversal */
func postOrder(node *TreeNode) {
if node == nil {
return
}
// Visit order: left subtree -> right subtree -> root
postOrder(node.Left)
postOrder(node.Right)
nums = append(nums, node.Val)
}
binary_tree_dfs.swift
/* Preorder traversal */
func preOrder(root: TreeNode?) {
guard let root = root else {
return
}
// Visit order: root -> 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 order: left subtree -> root -> 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 order: left subtree -> right subtree -> root
postOrder(root: root.left)
postOrder(root: root.right)
list.append(root.val)
}
binary_tree_dfs.js
/* Preorder traversal */
function preOrder(root) {
if (root === null) return;
// Visit order: root -> left subtree -> right subtree
list.push(root.val);
preOrder(root.left);
preOrder(root.right);
}
/* Inorder traversal */
function inOrder(root) {
if (root === null) return;
// Visit order: left subtree -> root -> right subtree
inOrder(root.left);
list.push(root.val);
inOrder(root.right);
}
/* Postorder traversal */
function postOrder(root) {
if (root === null) return;
// Visit order: left subtree -> right subtree -> root
postOrder(root.left);
postOrder(root.right);
list.push(root.val);
}
binary_tree_dfs.ts
/* Preorder traversal */
function preOrder(root: TreeNode | null): void {
if (root === null) {
return;
}
// Visit order: root -> 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 order: left subtree -> root -> 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 order: left subtree -> right subtree -> root
postOrder(root.left);
postOrder(root.right);
list.push(root.val);
}
binary_tree_dfs.dart
/* Preorder traversal */
void preOrder(TreeNode? node) {
if (node == null) return;
// Visit order: root -> 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 order: left subtree -> root -> right subtree
inOrder(node.left);
list.add(node.val);
inOrder(node.right);
}
/* Postorder traversal */
void postOrder(TreeNode? node) {
if (node == null) return;
// Visit order: left subtree -> right subtree -> root
postOrder(node.left);
postOrder(node.right);
list.add(node.val);
}
binary_tree_dfs.rs
/* Preorder traversal */
fn pre_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
let mut result = vec![];
fn dfs(root: Option<&Rc<RefCell<TreeNode>>>, res: &mut Vec<i32>) {
if let Some(node) = root {
// Visit order: root -> 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<RefCell<TreeNode>>>) -> Vec<i32> {
let mut result = vec![];
fn dfs(root: Option<&Rc<RefCell<TreeNode>>>, res: &mut Vec<i32>) {
if let Some(node) = root {
// Visit order: left subtree -> root -> 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<RefCell<TreeNode>>>) -> Vec<i32> {
let mut result = vec![];
fn dfs(root: Option<&Rc<RefCell<TreeNode>>>, res: &mut Vec<i32>) {
if let Some(node) = root {
// Visit order: left subtree -> right subtree -> root
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
}
binary_tree_dfs.c
/* Preorder traversal */
void preOrder(TreeNode *root, int *size) {
if (root == NULL)
return;
// Visit order: root -> 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 order: left subtree -> root -> 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 order: left subtree -> right subtree -> root
postOrder(root->left, size);
postOrder(root->right, size);
arr[(*size)++] = root->val;
}
binary_tree_dfs.kt
/* Preorder traversal */
fun preOrder(root: TreeNode?) {
if (root == null) return
// Visit order: root -> 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 order: left subtree -> root -> right subtree
inOrder(root.left)
list.add(root._val)
inOrder(root.right)
}
/* Postorder traversal */
fun postOrder(root: TreeNode?) {
if (root == null) return
// Visit order: left subtree -> right subtree -> root
postOrder(root.left)
postOrder(root.right)
list.add(root._val)
}
binary_tree_dfs.rb
### Preorder traversal ###
def pre_order(root)
return if root.nil?
# Visit order: root -> left subtree -> right subtree
$res << root.val
pre_order(root.left)
pre_order(root.right)
end
### Inorder traversal ###
def in_order(root)
return if root.nil?
# Visit order: left subtree -> root -> right subtree
in_order(root.left)
$res << root.val
in_order(root.right)
end
### Postorder traversal ###
def post_order(root)
return if root.nil?
# Visit order: left subtree -> right subtree -> root
post_order(root.left)
post_order(root.right)
$res << root.val
end
Visualize execution
Tip
Depth-first search can also be implemented iteratively. Interested readers can explore this approach themselves.
Figure 7-11 shows the recursive process for preorder traversal, with two opposing phases: descending into calls and returning from them.
- Descending starts a new function call and visits the next node.
- Returning ends a function call, indicating that the current node has been fully processed.
Figure 7-11 The recursive process of preorder traversal
2. Complexity analysis¶
- Time complexity: \(O(n)\). Every node is visited once, taking \(O(n)\) time.
- Space complexity: \(O(n)\). In the worst case, the tree degenerates into a linked list, recursion reaches depth \(n\), and stack frames require \(O(n)\) space.