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---
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comments: true
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---
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# 7.2 二元樹走訪
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從物理結構的角度來看,樹是一種基於鏈結串列的資料結構,因此其走訪方式是透過指標逐個訪問節點。然而,樹是一種非線性資料結構,這使得走訪樹比走訪鏈結串列更加複雜,需要藉助搜尋演算法來實現。
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二元樹常見的走訪方式包括層序走訪、前序走訪、中序走訪和後序走訪等。
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## 7.2.1 層序走訪
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如圖 7-9 所示,<u>層序走訪(level-order traversal)</u>從頂部到底部逐層走訪二元樹,並在每一層按照從左到右的順序訪問節點。
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層序走訪本質上屬於<u>廣度優先走訪(breadth-first traversal)</u>,也稱<u>廣度優先搜尋(breadth-first search, BFS)</u>,它體現了一種“一圈一圈向外擴展”的逐層走訪方式。
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![二元樹的層序走訪](binary_tree_traversal.assets/binary_tree_bfs.png){ class="animation-figure" }
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<p align="center"> 圖 7-9 二元樹的層序走訪 </p>
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### 1. 程式碼實現
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廣度優先走訪通常藉助“佇列”來實現。佇列遵循“先進先出”的規則,而廣度優先走訪則遵循“逐層推進”的規則,兩者背後的思想是一致的。實現程式碼如下:
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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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"""層序走訪"""
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# 初始化佇列,加入根節點
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queue: deque[TreeNode] = deque()
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queue.append(root)
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# 初始化一個串列,用於儲存走訪序列
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res = []
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while queue:
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node: TreeNode = queue.popleft() # 隊列出隊
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res.append(node.val) # 儲存節點值
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if node.left is not None:
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queue.append(node.left) # 左子節點入列
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if node.right is not None:
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queue.append(node.right) # 右子節點入列
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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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/* 層序走訪 */
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vector<int> levelOrder(TreeNode *root) {
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// 初始化佇列,加入根節點
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queue<TreeNode *> queue;
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queue.push(root);
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// 初始化一個串列,用於儲存走訪序列
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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(); // 隊列出隊
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vec.push_back(node->val); // 儲存節點值
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if (node->left != nullptr)
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queue.push(node->left); // 左子節點入列
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if (node->right != nullptr)
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queue.push(node->right); // 右子節點入列
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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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/* 層序走訪 */
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List<Integer> levelOrder(TreeNode root) {
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// 初始化佇列,加入根節點
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Queue<TreeNode> queue = new LinkedList<>();
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queue.add(root);
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// 初始化一個串列,用於儲存走訪序列
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List<Integer> list = new ArrayList<>();
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while (!queue.isEmpty()) {
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TreeNode node = queue.poll(); // 隊列出隊
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list.add(node.val); // 儲存節點值
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if (node.left != null)
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queue.offer(node.left); // 左子節點入列
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if (node.right != null)
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queue.offer(node.right); // 右子節點入列
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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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/* 層序走訪 */
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List<int> LevelOrder(TreeNode root) {
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// 初始化佇列,加入根節點
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Queue<TreeNode> queue = new();
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queue.Enqueue(root);
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// 初始化一個串列,用於儲存走訪序列
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List<int> list = [];
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while (queue.Count != 0) {
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TreeNode node = queue.Dequeue(); // 隊列出隊
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list.Add(node.val!.Value); // 儲存節點值
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if (node.left != null)
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queue.Enqueue(node.left); // 左子節點入列
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if (node.right != null)
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queue.Enqueue(node.right); // 右子節點入列
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}
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return list;
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}
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```
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=== "Go"
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```go title="binary_tree_bfs.go"
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/* 層序走訪 */
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func levelOrder(root *TreeNode) []any {
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// 初始化佇列,加入根節點
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queue := list.New()
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queue.PushBack(root)
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// 初始化一個切片,用於儲存走訪序列
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nums := make([]any, 0)
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for queue.Len() > 0 {
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// 隊列出隊
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node := queue.Remove(queue.Front()).(*TreeNode)
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// 儲存節點值
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nums = append(nums, node.Val)
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if node.Left != nil {
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// 左子節點入列
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queue.PushBack(node.Left)
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}
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if node.Right != nil {
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// 右子節點入列
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queue.PushBack(node.Right)
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}
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}
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return nums
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}
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```
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=== "Swift"
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```swift title="binary_tree_bfs.swift"
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/* 層序走訪 */
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func levelOrder(root: TreeNode) -> [Int] {
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// 初始化佇列,加入根節點
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var queue: [TreeNode] = [root]
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// 初始化一個串列,用於儲存走訪序列
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var list: [Int] = []
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while !queue.isEmpty {
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let node = queue.removeFirst() // 隊列出隊
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list.append(node.val) // 儲存節點值
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if let left = node.left {
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queue.append(left) // 左子節點入列
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}
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if let right = node.right {
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queue.append(right) // 右子節點入列
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}
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}
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return list
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}
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```
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=== "JS"
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```javascript title="binary_tree_bfs.js"
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/* 層序走訪 */
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function levelOrder(root) {
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// 初始化佇列,加入根節點
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const queue = [root];
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// 初始化一個串列,用於儲存走訪序列
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const list = [];
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while (queue.length) {
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let node = queue.shift(); // 隊列出隊
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list.push(node.val); // 儲存節點值
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if (node.left) queue.push(node.left); // 左子節點入列
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if (node.right) queue.push(node.right); // 右子節點入列
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}
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return list;
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}
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```
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=== "TS"
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```typescript title="binary_tree_bfs.ts"
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/* 層序走訪 */
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function levelOrder(root: TreeNode | null): number[] {
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// 初始化佇列,加入根節點
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const queue = [root];
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// 初始化一個串列,用於儲存走訪序列
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const list: number[] = [];
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while (queue.length) {
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let node = queue.shift() as TreeNode; // 隊列出隊
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list.push(node.val); // 儲存節點值
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if (node.left) {
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queue.push(node.left); // 左子節點入列
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}
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if (node.right) {
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queue.push(node.right); // 右子節點入列
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}
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}
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return list;
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}
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```
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=== "Dart"
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```dart title="binary_tree_bfs.dart"
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/* 層序走訪 */
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List<int> levelOrder(TreeNode? root) {
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// 初始化佇列,加入根節點
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Queue<TreeNode?> queue = Queue();
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queue.add(root);
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// 初始化一個串列,用於儲存走訪序列
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List<int> res = [];
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while (queue.isNotEmpty) {
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TreeNode? node = queue.removeFirst(); // 隊列出隊
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res.add(node!.val); // 儲存節點值
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if (node.left != null) queue.add(node.left); // 左子節點入列
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if (node.right != null) queue.add(node.right); // 右子節點入列
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}
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return res;
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}
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```
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=== "Rust"
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```rust title="binary_tree_bfs.rs"
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/* 層序走訪 */
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fn level_order(root: &Rc<RefCell<TreeNode>>) -> Vec<i32> {
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// 初始化佇列,加入根節點
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let mut que = VecDeque::new();
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que.push_back(root.clone());
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// 初始化一個串列,用於儲存走訪序列
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let mut vec = Vec::new();
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while let Some(node) = que.pop_front() {
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// 隊列出隊
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vec.push(node.borrow().val); // 儲存節點值
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if let Some(left) = node.borrow().left.as_ref() {
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que.push_back(left.clone()); // 左子節點入列
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}
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if let Some(right) = node.borrow().right.as_ref() {
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que.push_back(right.clone()); // 右子節點入列
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};
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}
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vec
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}
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```
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=== "C"
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```c title="binary_tree_bfs.c"
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/* 層序走訪 */
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int *levelOrder(TreeNode *root, int *size) {
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/* 輔助佇列 */
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int front, rear;
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int index, *arr;
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TreeNode *node;
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TreeNode **queue;
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/* 輔助佇列 */
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queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);
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// 佇列指標
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front = 0, rear = 0;
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// 加入根節點
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queue[rear++] = root;
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// 初始化一個串列,用於儲存走訪序列
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/* 輔助陣列 */
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arr = (int *)malloc(sizeof(int) * MAX_SIZE);
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// 陣列指標
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index = 0;
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while (front < rear) {
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// 隊列出隊
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node = queue[front++];
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// 儲存節點值
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arr[index++] = node->val;
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if (node->left != NULL) {
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// 左子節點入列
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queue[rear++] = node->left;
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}
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if (node->right != NULL) {
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// 右子節點入列
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queue[rear++] = node->right;
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}
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}
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// 更新陣列長度的值
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*size = index;
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arr = realloc(arr, sizeof(int) * (*size));
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// 釋放輔助陣列空間
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free(queue);
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return arr;
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}
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```
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=== "Kotlin"
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```kotlin title="binary_tree_bfs.kt"
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/* 層序走訪 */
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fun levelOrder(root: TreeNode?): MutableList<Int> {
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// 初始化佇列,加入根節點
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|
|
val queue = LinkedList<TreeNode?>()
|
|
|
|
|
queue.add(root)
|
|
|
|
|
// 初始化一個串列,用於儲存走訪序列
|
|
|
|
|
val list = mutableListOf<Int>()
|
|
|
|
|
while (queue.isNotEmpty()) {
|
|
|
|
|
val node = queue.poll() // 隊列出隊
|
|
|
|
|
list.add(node?._val!!) // 儲存節點值
|
|
|
|
|
if (node.left != null)
|
|
|
|
|
queue.offer(node.left) // 左子節點入列
|
|
|
|
|
if (node.right != null)
|
|
|
|
|
queue.offer(node.right) // 右子節點入列
|
|
|
|
|
}
|
|
|
|
|
return list
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Ruby"
|
|
|
|
|
|
|
|
|
|
```ruby title="binary_tree_bfs.rb"
|
|
|
|
|
[class]{}-[func]{level_order}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Zig"
|
|
|
|
|
|
|
|
|
|
```zig title="binary_tree_bfs.zig"
|
|
|
|
|
// 層序走訪
|
|
|
|
|
fn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {
|
|
|
|
|
// 初始化佇列,加入根節點
|
|
|
|
|
const L = std.TailQueue(*inc.TreeNode(T));
|
|
|
|
|
var queue = L{};
|
|
|
|
|
var root_node = try mem_allocator.create(L.Node);
|
|
|
|
|
root_node.data = root;
|
|
|
|
|
queue.append(root_node);
|
|
|
|
|
// 初始化一個串列,用於儲存走訪序列
|
|
|
|
|
var list = std.ArrayList(T).init(std.heap.page_allocator);
|
|
|
|
|
while (queue.len > 0) {
|
|
|
|
|
var queue_node = queue.popFirst().?; // 隊列出隊
|
|
|
|
|
var node = queue_node.data;
|
|
|
|
|
try list.append(node.val); // 儲存節點值
|
|
|
|
|
if (node.left != null) {
|
|
|
|
|
var tmp_node = try mem_allocator.create(L.Node);
|
|
|
|
|
tmp_node.data = node.left.?;
|
|
|
|
|
queue.append(tmp_node); // 左子節點入列
|
|
|
|
|
}
|
|
|
|
|
if (node.right != null) {
|
|
|
|
|
var tmp_node = try mem_allocator.create(L.Node);
|
|
|
|
|
tmp_node.data = node.right.?;
|
|
|
|
|
queue.append(tmp_node); // 右子節點入列
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
return list;
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
??? pythontutor "視覺化執行"
|
|
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|
|
|
|
|
|
|
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0Aclass%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D%2C%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%EF%BC%9A%E9%81%9E%E8%BF%B4%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E9%99%A3%E5%88%97%E9%95%B7%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%B0%8D%E6%87%89%E7%9A%84%E5%85%83%E7%B4%A0%E7%82%BA%20None%20%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%95%B6%E5%89%8D%E7%AF%80%E9%BB%9E%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%81%9E%E8%BF%B4%E6%A7%8B%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr%2C%200%29%0A%0A%0Adef%20level_order%28root%3A%20TreeNode%20%7C%20None%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22%E5%B1%A4%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BD%87%E5%88%97%EF%BC%8C%E5%8A%A0%E5%85%A5%E6%A0%B9%E7%AF%80%E9%BB%9E%0A%20%20%20%20queue%3A%20deque%5BTreeNode%5D%20%3D%20deque%28%29%0A%20%20%20%20queue.append%28root%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%B8%80%E5%80%8B%E4%B8%B2%E5%88%97%EF%BC%8C%E7%94%A8%E6%96%BC%E5%84%B2%E5%AD%98%E8%B5%B0%E8%A8%AA%E5%BA%8F%E5%88%97%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20while%20queue%3A%0A%20%20%20%20%20%20%20%20node%3A%20TreeNode%20%3D%20queue.popleft%28%29%20%20%23%20%E9%9A%8A%E5%88%97%E5%87%BA%E9%9A%8A%0A%20%20%20%20%20%20%20%20res.append%28node.val%29%20%20%23%20%E5%84%B2%E5%AD%98%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20if%20node.left%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.left%29%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%85%A5%E5%88%97%0A%20%20%20%20%20%20%20%20if%20node.right%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.right%29%20%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%85%A5%E5%88%97%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E9%80%99%E8%A3%A1%E8%97%89%E5%8A%A9%E4%BA%86%E4%B8%80%E5%80%8B%E5%BE%9E%E9%99%A3%E5%88%97%E7%9B%B4%E6%8E%A5%E7%94%9F%E6%88%90%E4%BA%8C%E5%85%83%E6%A8%B9%E7%9A%84%E5%87%BD%E5%BC%8F%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1%2C%202%2C%203%2C%204%2C%205%2C%206%2C%207%5D%29%0A%0A%20%20%20%20%23%20%E5%B1%A4%E5%BA%8F%E8%B5%B0%E8%A8%AA%0A%20%20%20%20res%20%3D%20level_order%28root%29%0A%20%20%20%20print%28%22%5Cn%E5%B1%A4%E5%BA%8F%E8%B5%B0%E8%A8%AA%E7%9A%84%E7%AF%80%E9%BB%9E%E5%88%97%E5%8D%B0%E5%BA%8F%E5%88%97%20%3D%20%22%2C%20res%29&codeDivHeigh
|
|
|
|
|
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0Aclass%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D%2C%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%EF%BC%9A%E9%81%9E%E8%BF%B4%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E9%99%A3%E5%88%97%E9%95%B7%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%B0%8D%E6%87%89%E7%9A%84%E5%85%83%E7%B4%A0%E7%82%BA%20None%20%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%95%B6%E5%89%8D%E7%AF%80%E9%BB%9E%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%81%9E%E8%BF%B4%E6%A7%8B%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr%2C%200%29%0A%0A%0Adef%20level_order%28root%3A%20TreeNode%20%7C%20None%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22%E5%B1%A4%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BD%87%E5%88%97%EF%BC%8C%E5%8A%A0%E5%85%A5%E6%A0%B9%E7%AF%80%E9%BB%9E%0A%20%20%20%20queue%3A%20deque%5BTreeNode%5D%20%3D%20deque%28%29%0A%20%20%20%20queue.append%28root%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%B8%80%E5%80%8B%E4%B8%B2%E5%88%97%EF%BC%8C%E7%94%A8%E6%96%BC%E5%84%B2%E5%AD%98%E8%B5%B0%E8%A8%AA%E5%BA%8F%E5%88%97%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20while%20queue%3A%0A%20%20%20%20%20%20%20%20node%3A%20TreeNode%20%3D%20queue.popleft%28%29%20%20%23%20%E9%9A%8A%E5%88%97%E5%87%BA%E9%9A%8A%0A%20%20%20%20%20%20%20%20res.append%28node.val%29%20%20%23%20%E5%84%B2%E5%AD%98%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20if%20node.left%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.left%29%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%85%A5%E5%88%97%0A%20%20%20%20%20%20%20%20if%20node.right%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.right%29%20%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%85%A5%E5%88%97%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E9%80%99%E8%A3%A1%E8%97%89%E5%8A%A9%E4%BA%86%E4%B8%80%E5%80%8B%E5%BE%9E%E9%99%A3%E5%88%97%E7%9B%B4%E6%8E%A5%E7%94%9F%E6%88%90%E4%BA%8C%E5%85%83%E6%A8%B9%E7%9A%84%E5%87%BD%E5%BC%8F%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1%2C%202%2C%203%2C%204%2C%205%2C%206%2C%207%5D%29%0A%0A%20%20%20%20%23%20%E5%B1%A4%E5%BA%8F%E8%B5%B0%E8%A8%AA%0A%20%20%20%20res%20%3D%20level_order%28root%29%0A%20%20%20%20print%28%22%5Cn%E5%B1%A4%E5%BA%8F%E8%B5%B0%E8%A8%AA%E7%9A%84%E7%AF%80%E9%BB%9E%E5%88%97%E5%8D%B0%E5%BA%8F%E5%88%97%20%3D%20%22%2C%20res%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&cu
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|
|
|
|
### 2. 複雜度分析
|
|
|
|
|
|
|
|
|
|
- **時間複雜度為 $O(n)$** :所有節點被訪問一次,使用 $O(n)$ 時間,其中 $n$ 為節點數量。
|
|
|
|
|
- **空間複雜度為 $O(n)$** :在最差情況下,即滿二元樹時,走訪到最底層之前,佇列中最多同時存在 $(n + 1) / 2$ 個節點,佔用 $O(n)$ 空間。
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|
|
|
|
|
## 7.2.2 前序、中序、後序走訪
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|
|
相應地,前序、中序和後序走訪都屬於<u>深度優先走訪(depth-first traversal)</u>,也稱<u>深度優先搜尋(depth-first search, DFS)</u>,它體現了一種“先走到盡頭,再回溯繼續”的走訪方式。
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|
|
圖 7-10 展示了對二元樹進行深度優先走訪的工作原理。**深度優先走訪就像是繞著整棵二元樹的外圍“走”一圈**,在每個節點都會遇到三個位置,分別對應前序走訪、中序走訪和後序走訪。
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|
|
![二元搜尋樹的前序、中序、後序走訪](binary_tree_traversal.assets/binary_tree_dfs.png){ class="animation-figure" }
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|
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|
|
<p align="center"> 圖 7-10 二元搜尋樹的前序、中序、後序走訪 </p>
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### 1. 程式碼實現
|
|
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深度優先搜尋通常基於遞迴實現:
|
|
|
|
|
|
|
|
|
|
=== "Python"
|
|
|
|
|
|
|
|
|
|
```python title="binary_tree_dfs.py"
|
|
|
|
|
def pre_order(root: TreeNode | None):
|
|
|
|
|
"""前序走訪"""
|
|
|
|
|
if root is None:
|
|
|
|
|
return
|
|
|
|
|
# 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
res.append(root.val)
|
|
|
|
|
pre_order(root=root.left)
|
|
|
|
|
pre_order(root=root.right)
|
|
|
|
|
|
|
|
|
|
def in_order(root: TreeNode | None):
|
|
|
|
|
"""中序走訪"""
|
|
|
|
|
if root is None:
|
|
|
|
|
return
|
|
|
|
|
# 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
in_order(root=root.left)
|
|
|
|
|
res.append(root.val)
|
|
|
|
|
in_order(root=root.right)
|
|
|
|
|
|
|
|
|
|
def post_order(root: TreeNode | None):
|
|
|
|
|
"""後序走訪"""
|
|
|
|
|
if root is None:
|
|
|
|
|
return
|
|
|
|
|
# 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
post_order(root=root.left)
|
|
|
|
|
post_order(root=root.right)
|
|
|
|
|
res.append(root.val)
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C++"
|
|
|
|
|
|
|
|
|
|
```cpp title="binary_tree_dfs.cpp"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
void preOrder(TreeNode *root) {
|
|
|
|
|
if (root == nullptr)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
vec.push_back(root->val);
|
|
|
|
|
preOrder(root->left);
|
|
|
|
|
preOrder(root->right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
void inOrder(TreeNode *root) {
|
|
|
|
|
if (root == nullptr)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root->left);
|
|
|
|
|
vec.push_back(root->val);
|
|
|
|
|
inOrder(root->right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
void postOrder(TreeNode *root) {
|
|
|
|
|
if (root == nullptr)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root->left);
|
|
|
|
|
postOrder(root->right);
|
|
|
|
|
vec.push_back(root->val);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Java"
|
|
|
|
|
|
|
|
|
|
```java title="binary_tree_dfs.java"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
void preOrder(TreeNode root) {
|
|
|
|
|
if (root == null)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.add(root.val);
|
|
|
|
|
preOrder(root.left);
|
|
|
|
|
preOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
void inOrder(TreeNode root) {
|
|
|
|
|
if (root == null)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root.left);
|
|
|
|
|
list.add(root.val);
|
|
|
|
|
inOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
void postOrder(TreeNode root) {
|
|
|
|
|
if (root == null)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root.left);
|
|
|
|
|
postOrder(root.right);
|
|
|
|
|
list.add(root.val);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
|
|
|
|
```csharp title="binary_tree_dfs.cs"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
void PreOrder(TreeNode? root) {
|
|
|
|
|
if (root == null) return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.Add(root.val!.Value);
|
|
|
|
|
PreOrder(root.left);
|
|
|
|
|
PreOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
void InOrder(TreeNode? root) {
|
|
|
|
|
if (root == null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
InOrder(root.left);
|
|
|
|
|
list.Add(root.val!.Value);
|
|
|
|
|
InOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
void PostOrder(TreeNode? root) {
|
|
|
|
|
if (root == null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
PostOrder(root.left);
|
|
|
|
|
PostOrder(root.right);
|
|
|
|
|
list.Add(root.val!.Value);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Go"
|
|
|
|
|
|
|
|
|
|
```go title="binary_tree_dfs.go"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
func preOrder(node *TreeNode) {
|
|
|
|
|
if node == nil {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
nums = append(nums, node.Val)
|
|
|
|
|
preOrder(node.Left)
|
|
|
|
|
preOrder(node.Right)
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
func inOrder(node *TreeNode) {
|
|
|
|
|
if node == nil {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(node.Left)
|
|
|
|
|
nums = append(nums, node.Val)
|
|
|
|
|
inOrder(node.Right)
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
func postOrder(node *TreeNode) {
|
|
|
|
|
if node == nil {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(node.Left)
|
|
|
|
|
postOrder(node.Right)
|
|
|
|
|
nums = append(nums, node.Val)
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Swift"
|
|
|
|
|
|
|
|
|
|
```swift title="binary_tree_dfs.swift"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
func preOrder(root: TreeNode?) {
|
|
|
|
|
guard let root = root else {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.append(root.val)
|
|
|
|
|
preOrder(root: root.left)
|
|
|
|
|
preOrder(root: root.right)
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
func inOrder(root: TreeNode?) {
|
|
|
|
|
guard let root = root else {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root: root.left)
|
|
|
|
|
list.append(root.val)
|
|
|
|
|
inOrder(root: root.right)
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
func postOrder(root: TreeNode?) {
|
|
|
|
|
guard let root = root else {
|
|
|
|
|
return
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root: root.left)
|
|
|
|
|
postOrder(root: root.right)
|
|
|
|
|
list.append(root.val)
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "JS"
|
|
|
|
|
|
|
|
|
|
```javascript title="binary_tree_dfs.js"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
function preOrder(root) {
|
|
|
|
|
if (root === null) return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.push(root.val);
|
|
|
|
|
preOrder(root.left);
|
|
|
|
|
preOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
function inOrder(root) {
|
|
|
|
|
if (root === null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root.left);
|
|
|
|
|
list.push(root.val);
|
|
|
|
|
inOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
function postOrder(root) {
|
|
|
|
|
if (root === null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root.left);
|
|
|
|
|
postOrder(root.right);
|
|
|
|
|
list.push(root.val);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "TS"
|
|
|
|
|
|
|
|
|
|
```typescript title="binary_tree_dfs.ts"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
function preOrder(root: TreeNode | null): void {
|
|
|
|
|
if (root === null) {
|
|
|
|
|
return;
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.push(root.val);
|
|
|
|
|
preOrder(root.left);
|
|
|
|
|
preOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
function inOrder(root: TreeNode | null): void {
|
|
|
|
|
if (root === null) {
|
|
|
|
|
return;
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root.left);
|
|
|
|
|
list.push(root.val);
|
|
|
|
|
inOrder(root.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
function postOrder(root: TreeNode | null): void {
|
|
|
|
|
if (root === null) {
|
|
|
|
|
return;
|
|
|
|
|
}
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root.left);
|
|
|
|
|
postOrder(root.right);
|
|
|
|
|
list.push(root.val);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Dart"
|
|
|
|
|
|
|
|
|
|
```dart title="binary_tree_dfs.dart"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
void preOrder(TreeNode? node) {
|
|
|
|
|
if (node == null) return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.add(node.val);
|
|
|
|
|
preOrder(node.left);
|
|
|
|
|
preOrder(node.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
void inOrder(TreeNode? node) {
|
|
|
|
|
if (node == null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(node.left);
|
|
|
|
|
list.add(node.val);
|
|
|
|
|
inOrder(node.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
void postOrder(TreeNode? node) {
|
|
|
|
|
if (node == null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(node.left);
|
|
|
|
|
postOrder(node.right);
|
|
|
|
|
list.add(node.val);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Rust"
|
|
|
|
|
|
|
|
|
|
```rust title="binary_tree_dfs.rs"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
fn pre_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
|
|
|
|
|
let mut result = vec![];
|
|
|
|
|
|
|
|
|
|
if let Some(node) = root {
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
result.push(node.borrow().val);
|
|
|
|
|
result.extend(pre_order(node.borrow().left.as_ref()));
|
|
|
|
|
result.extend(pre_order(node.borrow().right.as_ref()));
|
|
|
|
|
}
|
|
|
|
|
result
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
fn in_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
|
|
|
|
|
let mut result = vec![];
|
|
|
|
|
|
|
|
|
|
if let Some(node) = root {
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
result.extend(in_order(node.borrow().left.as_ref()));
|
|
|
|
|
result.push(node.borrow().val);
|
|
|
|
|
result.extend(in_order(node.borrow().right.as_ref()));
|
|
|
|
|
}
|
|
|
|
|
result
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
fn post_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
|
|
|
|
|
let mut result = vec![];
|
|
|
|
|
|
|
|
|
|
if let Some(node) = root {
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
result.extend(post_order(node.borrow().left.as_ref()));
|
|
|
|
|
result.extend(post_order(node.borrow().right.as_ref()));
|
|
|
|
|
result.push(node.borrow().val);
|
|
|
|
|
}
|
|
|
|
|
result
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
|
|
|
|
```c title="binary_tree_dfs.c"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
void preOrder(TreeNode *root, int *size) {
|
|
|
|
|
if (root == NULL)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
arr[(*size)++] = root->val;
|
|
|
|
|
preOrder(root->left, size);
|
|
|
|
|
preOrder(root->right, size);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
void inOrder(TreeNode *root, int *size) {
|
|
|
|
|
if (root == NULL)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root->left, size);
|
|
|
|
|
arr[(*size)++] = root->val;
|
|
|
|
|
inOrder(root->right, size);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
void postOrder(TreeNode *root, int *size) {
|
|
|
|
|
if (root == NULL)
|
|
|
|
|
return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root->left, size);
|
|
|
|
|
postOrder(root->right, size);
|
|
|
|
|
arr[(*size)++] = root->val;
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
|
|
|
|
```kotlin title="binary_tree_dfs.kt"
|
|
|
|
|
/* 前序走訪 */
|
|
|
|
|
fun preOrder(root: TreeNode?) {
|
|
|
|
|
if (root == null) return
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
list.add(root._val)
|
|
|
|
|
preOrder(root.left)
|
|
|
|
|
preOrder(root.right)
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 中序走訪 */
|
|
|
|
|
fun inOrder(root: TreeNode?) {
|
|
|
|
|
if (root == null) return
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
inOrder(root.left)
|
|
|
|
|
list.add(root._val)
|
|
|
|
|
inOrder(root.right)
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 後序走訪 */
|
|
|
|
|
fun postOrder(root: TreeNode?) {
|
|
|
|
|
if (root == null) return
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
postOrder(root.left)
|
|
|
|
|
postOrder(root.right)
|
|
|
|
|
list.add(root._val)
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
=== "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"
|
|
|
|
|
// 前序走訪
|
|
|
|
|
fn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {
|
|
|
|
|
if (root == null) return;
|
|
|
|
|
// 訪問優先順序:根節點 -> 左子樹 -> 右子樹
|
|
|
|
|
try list.append(root.?.val);
|
|
|
|
|
try preOrder(T, root.?.left);
|
|
|
|
|
try preOrder(T, root.?.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// 中序走訪
|
|
|
|
|
fn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {
|
|
|
|
|
if (root == null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 根節點 -> 右子樹
|
|
|
|
|
try inOrder(T, root.?.left);
|
|
|
|
|
try list.append(root.?.val);
|
|
|
|
|
try inOrder(T, root.?.right);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// 後序走訪
|
|
|
|
|
fn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {
|
|
|
|
|
if (root == null) return;
|
|
|
|
|
// 訪問優先順序:左子樹 -> 右子樹 -> 根節點
|
|
|
|
|
try postOrder(T, root.?.left);
|
|
|
|
|
try postOrder(T, root.?.right);
|
|
|
|
|
try list.append(root.?.val);
|
|
|
|
|
}
|
|
|
|
|
```
|
|
|
|
|
|
|
|
|
|
??? pythontutor "視覺化執行"
|
|
|
|
|
|
|
|
|
|
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D%2C%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%EF%BC%9A%E9%81%9E%E8%BF%B4%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E9%99%A3%E5%88%97%E9%95%B7%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%B0%8D%E6%87%89%E7%9A%84%E5%85%83%E7%B4%A0%E7%82%BA%20None%20%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%95%B6%E5%89%8D%E7%AF%80%E9%BB%9E%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%81%9E%E8%BF%B4%E6%A7%8B%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr%2C%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E8%A8%AA%E5%95%8F%E5%84%AA%E5%85%88%E9%A0%86%E5%BA%8F%EF%BC%9A%E6%A0%B9%E7%AF%80%E9%BB%9E%20-%3E%20%E5%B7%A6%E5%AD%90%E6%A8%B9%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20res.append%28root.val%29%0A%20%20%20%20pre_order%28root%3Droot.left%29%0A%20%20%20%20pre_order%28root%3Droot.right%29%0A%0Adef%20in_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22%E4%B8%AD%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E8%A8%AA%E5%95%8F%E5%84%AA%E5%85%88%E9%A0%86%E5%BA%8F%EF%BC%9A%E5%B7%A6%E5%AD%90%E6%A8%B9%20-%3E%20%E6%A0%B9%E7%AF%80%E9%BB%9E%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20in_order%28root%3Droot.left%29%0A%20%20%20%20res.append%28root.val%29%0A%20%20%20%20in_order%28root%3Droot.right%29%0A%0Adef%20post_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22%E5%BE%8C%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E8%A8%AA%E5%95%8F%E5%84%AA%E5%85%88%E9%A0%86%E5%BA%8F%EF%BC%9A%E5%B7%A6%E5%AD%90%E6%A8%B9%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A8%B9%20-%3E%20%E6%A0%B9%E7%AF%80%E9%BB%9E%0A%20%20%20%20post_order%28root%3Droot.left%29%0A%20%20%20%20post_order%28root%3Droot.right%29%0A%20%20%20%20res.append%28root.val%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E9%80%99%E8%A3%A1%E8%97%89%E5%8A%A9%E4%BA%86%E4%B8%80%E5%80%8B%E5%BE%9E%E9%99%A3%E5%88%97%E7%9B%B4%E6%8E%A5%E7%94%9F%E6%88%90%E4%BA%8C%E5%85%83%E6%A8%B9%E7%9A%84%E5%87%BD%E5%BC%8F%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1%2C%202%2C%203%2C%204%2C%205%2C%206%2C%207%5D%29%0A%0A%20%
|
|
|
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D%2C%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%EF%BC%9A%E9%81%9E%E8%BF%B4%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E9%99%A3%E5%88%97%E9%95%B7%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%B0%8D%E6%87%89%E7%9A%84%E5%85%83%E7%B4%A0%E7%82%BA%20None%20%EF%BC%8C%E5%89%87%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%95%B6%E5%89%8D%E7%AF%80%E9%BB%9E%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%81%9E%E8%BF%B4%E6%A7%8B%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr%2C%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%87%E4%B8%B2%E5%88%97%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E7%82%BA%E4%BA%8C%E5%85%83%E6%A8%B9%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr%2C%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E8%A8%AA%E5%95%8F%E5%84%AA%E5%85%88%E9%A0%86%E5%BA%8F%EF%BC%9A%E6%A0%B9%E7%AF%80%E9%BB%9E%20-%3E%20%E5%B7%A6%E5%AD%90%E6%A8%B9%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20res.append%28root.val%29%0A%20%20%20%20pre_order%28root%3Droot.left%29%0A%20%20%20%20pre_order%28root%3Droot.right%29%0A%0Adef%20in_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22%E4%B8%AD%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E8%A8%AA%E5%95%8F%E5%84%AA%E5%85%88%E9%A0%86%E5%BA%8F%EF%BC%9A%E5%B7%A6%E5%AD%90%E6%A8%B9%20-%3E%20%E6%A0%B9%E7%AF%80%E9%BB%9E%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A8%B9%0A%20%20%20%20in_order%28root%3Droot.left%29%0A%20%20%20%20res.append%28root.val%29%0A%20%20%20%20in_order%28root%3Droot.right%29%0A%0Adef%20post_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22%E5%BE%8C%E5%BA%8F%E8%B5%B0%E8%A8%AA%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E8%A8%AA%E5%95%8F%E5%84%AA%E5%85%88%E9%A0%86%E5%BA%8F%EF%BC%9A%E5%B7%A6%E5%AD%90%E6%A8%B9%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A8%B9%20-%3E%20%E6%A0%B9%E7%AF%80%E9%BB%9E%0A%20%20%20%20post_order%28root%3Droot.left%29%0A%20%20%20%20post_order%28root%3Droot.right%29%0A%20%20%20%20res.append%28root.val%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E9%80%99%E8%A3%A1%E8%97%89%E5%8A%A9%E4%BA%86%E4%B8%80%E5%80%8B%E5%BE%9E%E9%99%A3%E5%88%97%E7%9B%B4%E6%8E%A5%E7%94%9F%E6%88%90%E4%BA%8C%E5%85%83%E6%A8%B9%E7%9A%84%E5%87%BD%E5%BC%8F%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1%2C%202%2C%203%2C%204%2C%205%2C%206%2C%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E8%B5%B0%
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!!! tip
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深度優先搜尋也可以基於迭代實現,有興趣的讀者可以自行研究。
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圖 7-11 展示了前序走訪二元樹的遞迴過程,其可分為“遞”和“迴”兩個逆向的部分。
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1. “遞”表示開啟新方法,程式在此過程中訪問下一個節點。
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2. “迴”表示函式返回,代表當前節點已經訪問完畢。
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=== "<1>"
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![前序走訪的遞迴過程](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"> 圖 7-11 前序走訪的遞迴過程 </p>
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### 2. 複雜度分析
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- **時間複雜度為 $O(n)$** :所有節點被訪問一次,使用 $O(n)$ 時間。
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- **空間複雜度為 $O(n)$** :在最差情況下,即樹退化為鏈結串列時,遞迴深度達到 $n$ ,系統佔用 $O(n)$ 堆疊幀空間。
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