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

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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 breadth-first traversal, also known as breadth-first search (BFS), which proceeds outward level by level.

{ 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<int> levelOrder(TreeNode *root) {
    // Initialize queue, add root node
    queue<TreeNode *> queue;
    queue.push(root);
    // Initialize a list to save the traversal sequence
    vector<int> 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<Integer> levelOrder(TreeNode root) {
    // Initialize queue, add root node
    Queue<TreeNode> queue = new LinkedList<>();
    queue.add(root);
    // Initialize a list to save the traversal sequence
    List<Integer> 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<int> LevelOrder(TreeNode root) {
    // Initialize queue, add root node
    Queue<TreeNode> queue = new();
    queue.Enqueue(root);
    // Initialize a list to save the traversal sequence
    List<int> 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<int> levelOrder(TreeNode? root) {
  // Initialize queue, add root node
  Queue<TreeNode?> queue = Queue();
  queue.add(root);
  // Initialize a list to save the traversal sequence
  List<int> 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<RefCell<TreeNode>>) -> Vec<i32> {
    // 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<Int> {
    // Initialize queue, add root node
    val queue = LinkedList<TreeNode?>()
    queue.add(root)
    // Initialize a list to save the traversal sequence
    val list = mutableListOf<Int>()
    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.

{ 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<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 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<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 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<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 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>" { class="animation-figure" }

=== "<2>" { class="animation-figure" }

=== "<3>" { class="animation-figure" }

=== "<4>" { class="animation-figure" }

=== "<5>" { class="animation-figure" }

=== "<6>" { class="animation-figure" }

=== "<7>" { class="animation-figure" }

=== "<8>" { class="animation-figure" }

=== "<9>" { class="animation-figure" }

=== "<10>" { class="animation-figure" }

=== "<11>" { 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.