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en/docs/chapter_tree/binary_tree_traversal.md
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@@ -2,25 +2,25 @@
comments: true
---
# 7.2   Binary tree traversal
# 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
## 7.2.1   Level-Order Traversal
As shown in Figure 7-9, <u>level-order traversal</u> 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 a type of <u>breadth-first traversal</u>, also known as <u>breadth-first search (BFS)</u>, which embodies a "circumferentially outward expanding" layer-by-layer traversal method.
Level-order traversal is essentially <u>breadth-first traversal</u>, also known as <u>breadth-first search (BFS)</u>, which embodies a "expanding outward circle by circle" layer-by-layer traversal method.
![Level-order traversal of a binary tree](binary_tree_traversal.assets/binary_tree_bfs.png){ class="animation-figure" }
<p align="center"> Figure 7-9 &nbsp; Level-order traversal of a binary tree </p>
### 1. &nbsp; Code implementation
### 1. &nbsp; 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:
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"
@@ -30,15 +30,15 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
# Initialize queue, add root node
queue: deque[TreeNode] = deque()
queue.append(root)
# Initialize a list to store the traversal sequence
# Initialize a list to save the traversal sequence
res = []
while queue:
node: TreeNode = queue.popleft() # Queue dequeues
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 enqueues
queue.append(node.left) # Left child node enqueue
if node.right is not None:
queue.append(node.right) # Right child node enqueues
queue.append(node.right) # Right child node enqueue
return res
```
@@ -50,16 +50,16 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
// Initialize queue, add root node
queue<TreeNode *> queue;
queue.push(root);
// Initialize a list to store the traversal sequence
// Initialize a list to save the traversal sequence
vector<int> vec;
while (!queue.empty()) {
TreeNode *node = queue.front();
queue.pop(); // Queue dequeues
queue.pop(); // Dequeue
vec.push_back(node->val); // Save node value
if (node->left != nullptr)
queue.push(node->left); // Left child node enqueues
queue.push(node->left); // Left child node enqueue
if (node->right != nullptr)
queue.push(node->right); // Right child node enqueues
queue.push(node->right); // Right child node enqueue
}
return vec;
}
@@ -73,15 +73,15 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
// Initialize queue, add root node
Queue<TreeNode> queue = new LinkedList<>();
queue.add(root);
// Initialize a list to store the traversal sequence
// Initialize a list to save the traversal sequence
List<Integer> list = new ArrayList<>();
while (!queue.isEmpty()) {
TreeNode node = queue.poll(); // Queue dequeues
TreeNode node = queue.poll(); // Dequeue
list.add(node.val); // Save node value
if (node.left != null)
queue.offer(node.left); // Left child node enqueues
queue.offer(node.left); // Left child node enqueue
if (node.right != null)
queue.offer(node.right); // Right child node enqueues
queue.offer(node.right); // Right child node enqueue
}
return list;
}
@@ -90,85 +90,266 @@ Breadth-first traversal is usually implemented with the help of a "queue". The q
=== "C#"
```csharp title="binary_tree_bfs.cs"
[class]{binary_tree_bfs}-[func]{LevelOrder}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{level_order}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{levelOrder}
/* 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"
[class]{}-[func]{level_order}
### 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
```
=== "Zig"
### 2. &nbsp; Complexity Analysis
```zig title="binary_tree_bfs.zig"
[class]{}-[func]{levelOrder}
```
- **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.
### 2. &nbsp; Complexity analysis
## 7.2.2 &nbsp; Preorder, Inorder, and Postorder Traversal
- **Time complexity is $O(n)$**: All nodes are visited once, taking $O(n)$ 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 can contain at most $(n + 1) / 2$ nodes simultaneously, occupying $O(n)$ space.
Correspondingly, preorder, inorder, and postorder traversals all belong to <u>depth-first traversal</u>, also known as <u>depth-first search (DFS)</u>, which embodies a "first go to the end, then backtrack and continue" traversal method.
## 7.2.2 &nbsp; Preorder, in-order, and post-order traversal
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.
Correspondingly, pre-order, in-order, and post-order traversal all belong to <u>depth-first traversal</u>, also known as <u>depth-first search (DFS)</u>, which embodies a "proceed to the end first, then backtrack and continue" traversal method.
![Preorder, inorder, and postorder traversal of a binary tree](binary_tree_traversal.assets/binary_tree_dfs.png){ class="animation-figure" }
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 entire binary tree**, encountering three positions at each node, corresponding to pre-order, in-order, and post-order traversal.
<p align="center"> Figure 7-10 &nbsp; Preorder, inorder, and postorder traversal of a binary tree </p>
![Preorder, in-order, and post-order traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png){ class="animation-figure" }
<p align="center"> Figure 7-10 &nbsp; Preorder, in-order, and post-order traversal of a binary search tree </p>
### 1. &nbsp; Code implementation
### 1. &nbsp; Code Implementation
Depth-first search is usually implemented based on recursion:
@@ -176,7 +357,7 @@ Depth-first search is usually implemented based on recursion:
```python title="binary_tree_dfs.py"
def pre_order(root: TreeNode | None):
"""Pre-order traversal"""
"""Preorder traversal"""
if root is None:
return
# Visit priority: root node -> left subtree -> right subtree
@@ -185,7 +366,7 @@ Depth-first search is usually implemented based on recursion:
pre_order(root=root.right)
def in_order(root: TreeNode | None):
"""In-order traversal"""
"""Inorder traversal"""
if root is None:
return
# Visit priority: left subtree -> root node -> right subtree
@@ -194,7 +375,7 @@ Depth-first search is usually implemented based on recursion:
in_order(root=root.right)
def post_order(root: TreeNode | None):
"""Post-order traversal"""
"""Postorder traversal"""
if root is None:
return
# Visit priority: left subtree -> right subtree -> root node
@@ -206,7 +387,7 @@ Depth-first search is usually implemented based on recursion:
=== "C++"
```cpp title="binary_tree_dfs.cpp"
/* Pre-order traversal */
/* Preorder traversal */
void preOrder(TreeNode *root) {
if (root == nullptr)
return;
@@ -216,7 +397,7 @@ Depth-first search is usually implemented based on recursion:
preOrder(root->right);
}
/* In-order traversal */
/* Inorder traversal */
void inOrder(TreeNode *root) {
if (root == nullptr)
return;
@@ -226,7 +407,7 @@ Depth-first search is usually implemented based on recursion:
inOrder(root->right);
}
/* Post-order traversal */
/* Postorder traversal */
void postOrder(TreeNode *root) {
if (root == nullptr)
return;
@@ -240,7 +421,7 @@ Depth-first search is usually implemented based on recursion:
=== "Java"
```java title="binary_tree_dfs.java"
/* Pre-order traversal */
/* Preorder traversal */
void preOrder(TreeNode root) {
if (root == null)
return;
@@ -250,7 +431,7 @@ Depth-first search is usually implemented based on recursion:
preOrder(root.right);
}
/* In-order traversal */
/* Inorder traversal */
void inOrder(TreeNode root) {
if (root == null)
return;
@@ -260,7 +441,7 @@ Depth-first search is usually implemented based on recursion:
inOrder(root.right);
}
/* Post-order traversal */
/* Postorder traversal */
void postOrder(TreeNode root) {
if (root == null)
return;
@@ -274,124 +455,376 @@ Depth-first search is usually implemented based on recursion:
=== "C#"
```csharp title="binary_tree_dfs.cs"
[class]{binary_tree_dfs}-[func]{PreOrder}
/* 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);
}
[class]{binary_tree_dfs}-[func]{InOrder}
/* 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);
}
[class]{binary_tree_dfs}-[func]{PostOrder}
/* 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"
[class]{}-[func]{preOrder}
/* 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)
}
[class]{}-[func]{inOrder}
/* 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)
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{preOrder}
/* 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)
}
[class]{}-[func]{inOrder}
/* 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)
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{preOrder}
/* 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);
}
[class]{}-[func]{inOrder}
/* 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);
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{preOrder}
/* 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);
}
[class]{}-[func]{inOrder}
/* 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);
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{preOrder}
/* 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);
}
[class]{}-[func]{inOrder}
/* 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);
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{pre_order}
/* Preorder traversal */
fn pre_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
let mut result = vec![];
[class]{}-[func]{in_order}
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);
[class]{}-[func]{post_order}
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"
[class]{}-[func]{preOrder}
/* 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);
}
[class]{}-[func]{inOrder}
/* 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);
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{preOrder}
/* 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)
}
[class]{}-[func]{inOrder}
/* 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)
}
[class]{}-[func]{postOrder}
/* 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"
[class]{}-[func]{pre_order}
### Pre-order traversal ###
def pre_order(root)
return if root.nil?
[class]{}-[func]{in_order}
# Visit priority: root node -> left subtree -> right subtree
$res << root.val
pre_order(root.left)
pre_order(root.right)
end
[class]{}-[func]{post_order}
```
### In-order traversal ###
def in_order(root)
return if root.nil?
=== "Zig"
# Visit priority: left subtree -> root node -> right subtree
in_order(root.left)
$res << root.val
in_order(root.right)
end
```zig title="binary_tree_dfs.zig"
[class]{}-[func]{preOrder}
### Post-order traversal ###
def post_order(root)
return if root.nil?
[class]{}-[func]{inOrder}
[class]{}-[func]{postOrder}
# 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 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".
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. "Recursion" means opening a new method, where the program accesses the next node in this process.
2. "Return" means the function returns, indicating that the current node has been fully visited.
=== "<1>"
![The recursive process of pre-order traversal](binary_tree_traversal.assets/preorder_step1.png){ class="animation-figure" }
![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" }
@@ -423,9 +856,9 @@ Figure 7-11 shows the recursive process of pre-order traversal of a binary tree,
=== "<11>"
![preorder_step11](binary_tree_traversal.assets/preorder_step11.png){ class="animation-figure" }
<p align="center"> Figure 7-11 &nbsp; The recursive process of pre-order traversal </p>
<p align="center"> Figure 7-11 &nbsp; The recursive process of preorder traversal </p>
### 2. &nbsp; Complexity analysis
### 2. &nbsp; 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 $n$, the system occupies $O(n)$ stack frame space.
- **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.