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

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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, pre-order traversal, in-order traversal, and post-order 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.

{ 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]:
    """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
```

=== "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 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;
}
```

=== "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 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;
}
```

=== "C#"

```csharp title="binary_tree_bfs.cs"
[class]{binary_tree_bfs}-[func]{LevelOrder}
```

=== "Go"

```go title="binary_tree_bfs.go"
[class]{}-[func]{levelOrder}
```

=== "Swift"

```swift title="binary_tree_bfs.swift"
[class]{}-[func]{levelOrder}
```

=== "JS"

```javascript title="binary_tree_bfs.js"
[class]{}-[func]{levelOrder}
```

=== "TS"

```typescript title="binary_tree_bfs.ts"
[class]{}-[func]{levelOrder}
```

=== "Dart"

```dart title="binary_tree_bfs.dart"
[class]{}-[func]{levelOrder}
```

=== "Rust"

```rust title="binary_tree_bfs.rs"
[class]{}-[func]{level_order}
```

=== "C"

```c title="binary_tree_bfs.c"
[class]{}-[func]{levelOrder}
```

=== "Kotlin"

```kotlin title="binary_tree_bfs.kt"
[class]{}-[func]{levelOrder}
```

=== "Ruby"

```ruby title="binary_tree_bfs.rb"
[class]{}-[func]{level_order}
```

=== "Zig"

```zig title="binary_tree_bfs.zig"
[class]{}-[func]{levelOrder}
```

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, 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 perimeter of the entire binary tree, encountering three positions at each node, corresponding to pre-order traversal, in-order traversal, and post-order traversal.

{ class="animation-figure" }

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"

```python title="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)
```

=== "C++"

```cpp title="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);
}
```

=== "Java"

```java title="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);
}
```

=== "C#"

```csharp title="binary_tree_dfs.cs"
[class]{binary_tree_dfs}-[func]{PreOrder}

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

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

=== "Go"

```go title="binary_tree_dfs.go"
[class]{}-[func]{preOrder}

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

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

=== "Swift"

```swift title="binary_tree_dfs.swift"
[class]{}-[func]{preOrder}

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

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

=== "JS"

```javascript title="binary_tree_dfs.js"
[class]{}-[func]{preOrder}

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

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

=== "TS"

```typescript title="binary_tree_dfs.ts"
[class]{}-[func]{preOrder}

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

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

=== "Dart"

```dart title="binary_tree_dfs.dart"
[class]{}-[func]{preOrder}

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

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

=== "Rust"

```rust title="binary_tree_dfs.rs"
[class]{}-[func]{pre_order}

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

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

=== "C"

```c title="binary_tree_dfs.c"
[class]{}-[func]{preOrder}

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

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

=== "Kotlin"

```kotlin title="binary_tree_dfs.kt"
[class]{}-[func]{preOrder}

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

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

=== "Ruby"

```ruby title="binary_tree_dfs.rb"
[class]{}-[func]{pre_order}

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

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

=== "Zig"

```zig title="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>" { 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 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 degrades into a linked list, the recursion depth reaches n, the system occupies O(n) stack frame space.