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431 lines
12 KiB
Markdown
Executable file
431 lines
12 KiB
Markdown
Executable file
---
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comments: true
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---
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# 7.2 Binary tree traversal
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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.
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Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.
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## 7.2.1 Level-order traversal
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As shown in Figure 7-9, <u>level-order traversal</u> traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.
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Level-order traversal essentially belongs to <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.
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![Level-order traversal of a binary tree](binary_tree_traversal.assets/binary_tree_bfs.png){ class="animation-figure" }
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<p align="center"> Figure 7-9 Level-order traversal of a binary tree </p>
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### 1. Code implementation
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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:
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=== "Python"
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```python title="binary_tree_bfs.py"
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def level_order(root: TreeNode | None) -> list[int]:
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"""Level-order traversal"""
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# Initialize queue, add root node
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queue: deque[TreeNode] = deque()
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queue.append(root)
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# Initialize a list to store the traversal sequence
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res = []
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while queue:
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node: TreeNode = queue.popleft() # Queue dequeues
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res.append(node.val) # Save node value
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if node.left is not None:
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queue.append(node.left) # Left child node enqueues
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if node.right is not None:
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queue.append(node.right) # Right child node enqueues
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return res
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```
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=== "C++"
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```cpp title="binary_tree_bfs.cpp"
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/* Level-order traversal */
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vector<int> levelOrder(TreeNode *root) {
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// Initialize queue, add root node
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queue<TreeNode *> queue;
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queue.push(root);
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// Initialize a list to store the traversal sequence
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vector<int> vec;
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while (!queue.empty()) {
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TreeNode *node = queue.front();
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queue.pop(); // Queue dequeues
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vec.push_back(node->val); // Save node value
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if (node->left != nullptr)
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queue.push(node->left); // Left child node enqueues
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if (node->right != nullptr)
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queue.push(node->right); // Right child node enqueues
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}
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return vec;
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}
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```
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=== "Java"
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```java title="binary_tree_bfs.java"
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/* Level-order traversal */
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List<Integer> levelOrder(TreeNode root) {
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// Initialize queue, add root node
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Queue<TreeNode> queue = new LinkedList<>();
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queue.add(root);
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// Initialize a list to store the traversal sequence
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List<Integer> list = new ArrayList<>();
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while (!queue.isEmpty()) {
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TreeNode node = queue.poll(); // Queue dequeues
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list.add(node.val); // Save node value
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if (node.left != null)
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queue.offer(node.left); // Left child node enqueues
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if (node.right != null)
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queue.offer(node.right); // Right child node enqueues
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}
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return list;
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}
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```
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=== "C#"
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```csharp title="binary_tree_bfs.cs"
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[class]{binary_tree_bfs}-[func]{LevelOrder}
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```
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=== "Go"
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```go title="binary_tree_bfs.go"
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[class]{}-[func]{levelOrder}
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```
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=== "Swift"
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```swift title="binary_tree_bfs.swift"
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[class]{}-[func]{levelOrder}
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```
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=== "JS"
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```javascript title="binary_tree_bfs.js"
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[class]{}-[func]{levelOrder}
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```
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=== "TS"
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```typescript title="binary_tree_bfs.ts"
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[class]{}-[func]{levelOrder}
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```
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=== "Dart"
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```dart title="binary_tree_bfs.dart"
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[class]{}-[func]{levelOrder}
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```
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=== "Rust"
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```rust title="binary_tree_bfs.rs"
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[class]{}-[func]{level_order}
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```
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=== "C"
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```c title="binary_tree_bfs.c"
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[class]{}-[func]{levelOrder}
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```
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=== "Kotlin"
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```kotlin title="binary_tree_bfs.kt"
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[class]{}-[func]{levelOrder}
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```
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=== "Ruby"
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```ruby title="binary_tree_bfs.rb"
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[class]{}-[func]{level_order}
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```
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=== "Zig"
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```zig title="binary_tree_bfs.zig"
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[class]{}-[func]{levelOrder}
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```
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### 2. Complexity analysis
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- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time, where $n$ is the number of nodes.
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- **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.
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## 7.2.2 Preorder, inorder, and postorder traversal
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Correspondingly, preorder, inorder, and postorder 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.
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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.
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![Preorder, inorder, and postorder traversal of a binary search tree](binary_tree_traversal.assets/binary_tree_dfs.png){ class="animation-figure" }
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<p align="center"> Figure 7-10 Preorder, inorder, and postorder traversal of a binary search tree </p>
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### 1. Code implementation
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Depth-first search is usually implemented based on recursion:
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=== "Python"
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```python title="binary_tree_dfs.py"
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def pre_order(root: TreeNode | None):
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"""Pre-order traversal"""
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if root is None:
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return
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# Visit priority: root node -> left subtree -> right subtree
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res.append(root.val)
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pre_order(root=root.left)
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pre_order(root=root.right)
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def in_order(root: TreeNode | None):
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"""In-order traversal"""
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if root is None:
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return
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# Visit priority: left subtree -> root node -> right subtree
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in_order(root=root.left)
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res.append(root.val)
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in_order(root=root.right)
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def post_order(root: TreeNode | None):
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"""Post-order traversal"""
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if root is None:
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return
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# Visit priority: left subtree -> right subtree -> root node
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post_order(root=root.left)
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post_order(root=root.right)
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res.append(root.val)
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```
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=== "C++"
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```cpp title="binary_tree_dfs.cpp"
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/* Pre-order traversal */
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void preOrder(TreeNode *root) {
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if (root == nullptr)
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return;
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// Visit priority: root node -> left subtree -> right subtree
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vec.push_back(root->val);
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preOrder(root->left);
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preOrder(root->right);
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}
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/* In-order traversal */
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void inOrder(TreeNode *root) {
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if (root == nullptr)
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return;
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// Visit priority: left subtree -> root node -> right subtree
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inOrder(root->left);
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vec.push_back(root->val);
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inOrder(root->right);
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}
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/* Post-order traversal */
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void postOrder(TreeNode *root) {
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if (root == nullptr)
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return;
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// Visit priority: left subtree -> right subtree -> root node
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postOrder(root->left);
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postOrder(root->right);
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vec.push_back(root->val);
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}
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```
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=== "Java"
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```java title="binary_tree_dfs.java"
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/* Pre-order traversal */
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void preOrder(TreeNode root) {
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if (root == null)
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return;
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// Visit priority: root node -> left subtree -> right subtree
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list.add(root.val);
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preOrder(root.left);
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preOrder(root.right);
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}
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/* In-order traversal */
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void inOrder(TreeNode root) {
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if (root == null)
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return;
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// Visit priority: left subtree -> root node -> right subtree
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inOrder(root.left);
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list.add(root.val);
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inOrder(root.right);
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}
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/* Post-order traversal */
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void postOrder(TreeNode root) {
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if (root == null)
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return;
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// Visit priority: left subtree -> right subtree -> root node
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postOrder(root.left);
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postOrder(root.right);
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list.add(root.val);
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}
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```
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=== "C#"
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```csharp title="binary_tree_dfs.cs"
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[class]{binary_tree_dfs}-[func]{PreOrder}
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[class]{binary_tree_dfs}-[func]{InOrder}
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[class]{binary_tree_dfs}-[func]{PostOrder}
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```
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=== "Go"
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```go title="binary_tree_dfs.go"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Swift"
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```swift title="binary_tree_dfs.swift"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "JS"
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```javascript title="binary_tree_dfs.js"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "TS"
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```typescript title="binary_tree_dfs.ts"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Dart"
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```dart title="binary_tree_dfs.dart"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Rust"
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```rust title="binary_tree_dfs.rs"
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[class]{}-[func]{pre_order}
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[class]{}-[func]{in_order}
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[class]{}-[func]{post_order}
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```
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=== "C"
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```c title="binary_tree_dfs.c"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Kotlin"
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```kotlin title="binary_tree_dfs.kt"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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=== "Ruby"
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```ruby title="binary_tree_dfs.rb"
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[class]{}-[func]{pre_order}
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[class]{}-[func]{in_order}
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[class]{}-[func]{post_order}
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```
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=== "Zig"
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```zig title="binary_tree_dfs.zig"
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[class]{}-[func]{preOrder}
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[class]{}-[func]{inOrder}
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[class]{}-[func]{postOrder}
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```
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!!! tip
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Depth-first search can also be implemented based on iteration, interested readers can study this on their own.
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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".
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1. "Recursion" means starting a new method, the program accesses the next node in this process.
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2. "Return" means the function returns, indicating the current node has been fully accessed.
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=== "<1>"
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![The recursive process of preorder traversal](binary_tree_traversal.assets/preorder_step1.png){ class="animation-figure" }
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=== "<2>"
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![preorder_step2](binary_tree_traversal.assets/preorder_step2.png){ class="animation-figure" }
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=== "<3>"
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![preorder_step3](binary_tree_traversal.assets/preorder_step3.png){ class="animation-figure" }
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=== "<4>"
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![preorder_step4](binary_tree_traversal.assets/preorder_step4.png){ class="animation-figure" }
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=== "<5>"
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![preorder_step5](binary_tree_traversal.assets/preorder_step5.png){ class="animation-figure" }
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=== "<6>"
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![preorder_step6](binary_tree_traversal.assets/preorder_step6.png){ class="animation-figure" }
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=== "<7>"
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![preorder_step7](binary_tree_traversal.assets/preorder_step7.png){ class="animation-figure" }
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=== "<8>"
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![preorder_step8](binary_tree_traversal.assets/preorder_step8.png){ class="animation-figure" }
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=== "<9>"
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![preorder_step9](binary_tree_traversal.assets/preorder_step9.png){ class="animation-figure" }
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=== "<10>"
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![preorder_step10](binary_tree_traversal.assets/preorder_step10.png){ class="animation-figure" }
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=== "<11>"
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![preorder_step11](binary_tree_traversal.assets/preorder_step11.png){ class="animation-figure" }
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<p align="center"> Figure 7-11 The recursive process of preorder traversal </p>
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### 2. Complexity analysis
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- **Time complexity is $O(n)$**: All nodes are visited once, using $O(n)$ time.
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- **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.
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