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7.2 Binary tree traversal

From the perspective of physical structure, a tree is a data structure based on linked lists, hence its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms to achieve.

Common traversal methods for binary trees include level-order traversal, preorder traversal, inorder traversal, and postorder traversal, among others.

7.2.1 Level-order traversal

As shown in the Figure 7-9 , "level-order traversal" traverses the binary tree from top to bottom, layer by layer, and accesses nodes in each layer in a left-to-right order.

Level-order traversal essentially belongs to "breadth-first traversal", also known as "breadth-first search (BFS)", which embodies a "circumferentially outward expanding" layer-by-layer traversal method.

{ class="animation-figure" }

Figure 7-9 Level-order traversal of a binary tree

1. Code implementation

Breadth-first traversal is usually implemented with the help of a "queue". The queue follows the "first in, first out" rule, while breadth-first traversal follows the "layer-by-layer progression" rule, the underlying ideas of the two are consistent. The implementation code is as follows:

=== "Python"

```python title="binary_tree_bfs.py"
def level_order(root: TreeNode | None) -> list[int]:
    """层序遍历"""
    # 初始化队列，加入根节点
    queue: deque[TreeNode] = deque()
    queue.append(root)
    # 初始化一个列表，用于保存遍历序列
    res = []
    while queue:
        node: TreeNode = queue.popleft()  # 队列出队
        res.append(node.val)  # 保存节点值
        if node.left is not None:
            queue.append(node.left)  # 左子节点入队
        if node.right is not None:
            queue.append(node.right)  # 右子节点入队
    return res
```

=== "C++"

```cpp title="binary_tree_bfs.cpp"
/* 层序遍历 */
vector<int> levelOrder(TreeNode *root) {
    // 初始化队列，加入根节点
    queue<TreeNode *> queue;
    queue.push(root);
    // 初始化一个列表，用于保存遍历序列
    vector<int> vec;
    while (!queue.empty()) {
        TreeNode *node = queue.front();
        queue.pop();              // 队列出队
        vec.push_back(node->val); // 保存节点值
        if (node->left != nullptr)
            queue.push(node->left); // 左子节点入队
        if (node->right != nullptr)
            queue.push(node->right); // 右子节点入队
    }
    return vec;
}
```

=== "Java"

```java title="binary_tree_bfs.java"
/* 层序遍历 */
List<Integer> levelOrder(TreeNode root) {
    // 初始化队列，加入根节点
    Queue<TreeNode> queue = new LinkedList<>();
    queue.add(root);
    // 初始化一个列表，用于保存遍历序列
    List<Integer> list = new ArrayList<>();
    while (!queue.isEmpty()) {
        TreeNode node = queue.poll(); // 队列出队
        list.add(node.val);           // 保存节点值
        if (node.left != null)
            queue.offer(node.left);   // 左子节点入队
        if (node.right != null)
            queue.offer(node.right);  // 右子节点入队
    }
    return list;
}
```

=== "C#"

```csharp title="binary_tree_bfs.cs"
/* 层序遍历 */
List<int> LevelOrder(TreeNode root) {
    // 初始化队列，加入根节点
    Queue<TreeNode> queue = new();
    queue.Enqueue(root);
    // 初始化一个列表，用于保存遍历序列
    List<int> list = [];
    while (queue.Count != 0) {
        TreeNode node = queue.Dequeue(); // 队列出队
        list.Add(node.val!.Value);       // 保存节点值
        if (node.left != null)
            queue.Enqueue(node.left);    // 左子节点入队
        if (node.right != null)
            queue.Enqueue(node.right);   // 右子节点入队
    }
    return list;
}
```

=== "Go"

```go title="binary_tree_bfs.go"
/* 层序遍历 */
func levelOrder(root *TreeNode) []any {
    // 初始化队列，加入根节点
    queue := list.New()
    queue.PushBack(root)
    // 初始化一个切片，用于保存遍历序列
    nums := make([]any, 0)
    for queue.Len() > 0 {
        // 队列出队
        node := queue.Remove(queue.Front()).(*TreeNode)
        // 保存节点值
        nums = append(nums, node.Val)
        if node.Left != nil {
            // 左子节点入队
            queue.PushBack(node.Left)
        }
        if node.Right != nil {
            // 右子节点入队
            queue.PushBack(node.Right)
        }
    }
    return nums
}
```

=== "Swift"

```swift title="binary_tree_bfs.swift"
/* 层序遍历 */
func levelOrder(root: TreeNode) -> [Int] {
    // 初始化队列，加入根节点
    var queue: [TreeNode] = [root]
    // 初始化一个列表，用于保存遍历序列
    var list: [Int] = []
    while !queue.isEmpty {
        let node = queue.removeFirst() // 队列出队
        list.append(node.val) // 保存节点值
        if let left = node.left {
            queue.append(left) // 左子节点入队
        }
        if let right = node.right {
            queue.append(right) // 右子节点入队
        }
    }
    return list
}
```

=== "JS"

```javascript title="binary_tree_bfs.js"
/* 层序遍历 */
function levelOrder(root) {
    // 初始化队列，加入根节点
    const queue = [root];
    // 初始化一个列表，用于保存遍历序列
    const list = [];
    while (queue.length) {
        let node = queue.shift(); // 队列出队
        list.push(node.val); // 保存节点值
        if (node.left) queue.push(node.left); // 左子节点入队
        if (node.right) queue.push(node.right); // 右子节点入队
    }
    return list;
}
```

=== "TS"

```typescript title="binary_tree_bfs.ts"
/* 层序遍历 */
function levelOrder(root: TreeNode | null): number[] {
    // 初始化队列，加入根节点
    const queue = [root];
    // 初始化一个列表，用于保存遍历序列
    const list: number[] = [];
    while (queue.length) {
        let node = queue.shift() as TreeNode; // 队列出队
        list.push(node.val); // 保存节点值
        if (node.left) {
            queue.push(node.left); // 左子节点入队
        }
        if (node.right) {
            queue.push(node.right); // 右子节点入队
        }
    }
    return list;
}
```

=== "Dart"

```dart title="binary_tree_bfs.dart"
/* 层序遍历 */
List<int> levelOrder(TreeNode? root) {
  // 初始化队列，加入根节点
  Queue<TreeNode?> queue = Queue();
  queue.add(root);
  // 初始化一个列表，用于保存遍历序列
  List<int> res = [];
  while (queue.isNotEmpty) {
    TreeNode? node = queue.removeFirst(); // 队列出队
    res.add(node!.val); // 保存节点值
    if (node.left != null) queue.add(node.left); // 左子节点入队
    if (node.right != null) queue.add(node.right); // 右子节点入队
  }
  return res;
}
```

=== "Rust"

```rust title="binary_tree_bfs.rs"
/* 层序遍历 */
fn level_order(root: &Rc<RefCell<TreeNode>>) -> Vec<i32> {
    // 初始化队列，加入根节点
    let mut que = VecDeque::new();
    que.push_back(root.clone());
    // 初始化一个列表，用于保存遍历序列
    let mut vec = Vec::new();

    while let Some(node) = que.pop_front() {
        // 队列出队
        vec.push(node.borrow().val); // 保存节点值
        if let Some(left) = node.borrow().left.as_ref() {
            que.push_back(left.clone()); // 左子节点入队
        }
        if let Some(right) = node.borrow().right.as_ref() {
            que.push_back(right.clone()); // 右子节点入队
        };
    }
    vec
}
```

=== "C"

```c title="binary_tree_bfs.c"
/* 层序遍历 */
int *levelOrder(TreeNode *root, int *size) {
    /* 辅助队列 */
    int front, rear;
    int index, *arr;
    TreeNode *node;
    TreeNode **queue;

    /* 辅助队列 */
    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);
    // 队列指针
    front = 0, rear = 0;
    // 加入根节点
    queue[rear++] = root;
    // 初始化一个列表，用于保存遍历序列
    /* 辅助数组 */
    arr = (int *)malloc(sizeof(int) * MAX_SIZE);
    // 数组指针
    index = 0;
    while (front < rear) {
        // 队列出队
        node = queue[front++];
        // 保存节点值
        arr[index++] = node->val;
        if (node->left != NULL) {
            // 左子节点入队
            queue[rear++] = node->left;
        }
        if (node->right != NULL) {
            // 右子节点入队
            queue[rear++] = node->right;
        }
    }
    // 更新数组长度的值
    *size = index;
    arr = realloc(arr, sizeof(int) * (*size));

    // 释放辅助数组空间
    free(queue);
    return arr;
}
```

=== "Kotlin"

```kotlin title="binary_tree_bfs.kt"
/* 层序遍历 */
fun levelOrder(root: TreeNode?): MutableList<Int> {
    // 初始化队列，加入根节点
    val queue = LinkedList<TreeNode?>()
    queue.add(root)
    // 初始化一个列表，用于保存遍历序列
    val list = mutableListOf<Int>()
    while (queue.isNotEmpty()) {
        val node = queue.poll()      // 队列出队
        list.add(node?._val!!)       // 保存节点值
        if (node.left != null)
            queue.offer(node.left)   // 左子节点入队
        if (node.right != null)
            queue.offer(node.right)  // 右子节点入队
    }
    return list
}
```

=== "Ruby"

```ruby title="binary_tree_bfs.rb"
### 层序遍历 ###
def level_order(root)
  # 初始化队列，加入根节点
  queue = [root]
  # 初始化一个列表，用于保存遍历序列
  res = []
  while !queue.empty?
    node = queue.shift # 队列出队
    res << node.val # 保存节点值
    queue << node.left unless node.left.nil? # 左子节点入队
    queue << node.right unless node.right.nil? # 右子节点入队
  end
  res
end
```

=== "Zig"

```zig title="binary_tree_bfs.zig"
// 层序遍历
fn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {
    // 初始化队列，加入根节点
    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); 
    // 初始化一个列表，用于保存遍历序列
    var list = std.ArrayList(T).init(std.heap.page_allocator);
    while (queue.len > 0) {
        var queue_node = queue.popFirst().?;    // 队列出队
        var node = queue_node.data;
        try list.append(node.val);              // 保存节点值
        if (node.left != null) {
            var tmp_node = try mem_allocator.create(L.Node);
            tmp_node.data = node.left.?;
            queue.append(tmp_node);             // 左子节点入队
        }
        if (node.right != null) {
            var tmp_node = try mem_allocator.create(L.Node);
            tmp_node.data = node.right.?;
            queue.append(tmp_node);             // 右子节点入队
        }        
    }
    return list;
}
```

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2. Complexity analysis

Time complexity is O(n): All nodes are visited once, using O(n) time, where n is the number of nodes.
Space complexity is O(n): In the worst case, i.e., a full binary tree, before traversing to the lowest level, the queue can contain at most (n + 1) / 2 nodes at the same time, occupying O(n) space.

7.2.2 Preorder, inorder, and postorder traversal

Correspondingly, preorder, inorder, and postorder traversal all belong to "depth-first traversal", also known as "depth-first search (DFS)", which embodies a "proceed to the end first, then backtrack and continue" traversal method.

The 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.

{ class="animation-figure" }

Figure 7-10 Preorder, inorder, and postorder traversal of a binary search tree

1. Code implementation

Depth-first search is usually implemented based on recursion:

=== "Python"

```python title="binary_tree_dfs.py"
def pre_order(root: TreeNode | None):
    """前序遍历"""
    if root is None:
        return
    # 访问优先级：根节点 -> 左子树 -> 右子树
    res.append(root.val)
    pre_order(root=root.left)
    pre_order(root=root.right)

def in_order(root: TreeNode | None):
    """中序遍历"""
    if root is None:
        return
    # 访问优先级：左子树 -> 根节点 -> 右子树
    in_order(root=root.left)
    res.append(root.val)
    in_order(root=root.right)

def post_order(root: TreeNode | None):
    """后序遍历"""
    if root is None:
        return
    # 访问优先级：左子树 -> 右子树 -> 根节点
    post_order(root=root.left)
    post_order(root=root.right)
    res.append(root.val)
```

=== "C++"

```cpp title="binary_tree_dfs.cpp"
/* 前序遍历 */
void preOrder(TreeNode *root) {
    if (root == nullptr)
        return;
    // 访问优先级：根节点 -> 左子树 -> 右子树
    vec.push_back(root->val);
    preOrder(root->left);
    preOrder(root->right);
}

/* 中序遍历 */
void inOrder(TreeNode *root) {
    if (root == nullptr)
        return;
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root->left);
    vec.push_back(root->val);
    inOrder(root->right);
}

/* 后序遍历 */
void postOrder(TreeNode *root) {
    if (root == nullptr)
        return;
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root->left);
    postOrder(root->right);
    vec.push_back(root->val);
}
```

=== "Java"

```java title="binary_tree_dfs.java"
/* 前序遍历 */
void preOrder(TreeNode root) {
    if (root == null)
        return;
    // 访问优先级：根节点 -> 左子树 -> 右子树
    list.add(root.val);
    preOrder(root.left);
    preOrder(root.right);
}

/* 中序遍历 */
void inOrder(TreeNode root) {
    if (root == null)
        return;
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root.left);
    list.add(root.val);
    inOrder(root.right);
}

/* 后序遍历 */
void postOrder(TreeNode root) {
    if (root == null)
        return;
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root.left);
    postOrder(root.right);
    list.add(root.val);
}
```

=== "C#"

```csharp title="binary_tree_dfs.cs"
/* 前序遍历 */
void PreOrder(TreeNode? root) {
    if (root == null) return;
    // 访问优先级：根节点 -> 左子树 -> 右子树
    list.Add(root.val!.Value);
    PreOrder(root.left);
    PreOrder(root.right);
}

/* 中序遍历 */
void InOrder(TreeNode? root) {
    if (root == null) return;
    // 访问优先级：左子树 -> 根节点 -> 右子树
    InOrder(root.left);
    list.Add(root.val!.Value);
    InOrder(root.right);
}

/* 后序遍历 */
void PostOrder(TreeNode? root) {
    if (root == null) return;
    // 访问优先级：左子树 -> 右子树 -> 根节点
    PostOrder(root.left);
    PostOrder(root.right);
    list.Add(root.val!.Value);
}
```

=== "Go"

```go title="binary_tree_dfs.go"
/* 前序遍历 */
func preOrder(node *TreeNode) {
    if node == nil {
        return
    }
    // 访问优先级：根节点 -> 左子树 -> 右子树
    nums = append(nums, node.Val)
    preOrder(node.Left)
    preOrder(node.Right)
}

/* 中序遍历 */
func inOrder(node *TreeNode) {
    if node == nil {
        return
    }
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(node.Left)
    nums = append(nums, node.Val)
    inOrder(node.Right)
}

/* 后序遍历 */
func postOrder(node *TreeNode) {
    if node == nil {
        return
    }
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(node.Left)
    postOrder(node.Right)
    nums = append(nums, node.Val)
}
```

=== "Swift"

```swift title="binary_tree_dfs.swift"
/* 前序遍历 */
func preOrder(root: TreeNode?) {
    guard let root = root else {
        return
    }
    // 访问优先级：根节点 -> 左子树 -> 右子树
    list.append(root.val)
    preOrder(root: root.left)
    preOrder(root: root.right)
}

/* 中序遍历 */
func inOrder(root: TreeNode?) {
    guard let root = root else {
        return
    }
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root: root.left)
    list.append(root.val)
    inOrder(root: root.right)
}

/* 后序遍历 */
func postOrder(root: TreeNode?) {
    guard let root = root else {
        return
    }
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root: root.left)
    postOrder(root: root.right)
    list.append(root.val)
}
```

=== "JS"

```javascript title="binary_tree_dfs.js"
/* 前序遍历 */
function preOrder(root) {
    if (root === null) return;
    // 访问优先级：根节点 -> 左子树 -> 右子树
    list.push(root.val);
    preOrder(root.left);
    preOrder(root.right);
}

/* 中序遍历 */
function inOrder(root) {
    if (root === null) return;
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root.left);
    list.push(root.val);
    inOrder(root.right);
}

/* 后序遍历 */
function postOrder(root) {
    if (root === null) return;
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root.left);
    postOrder(root.right);
    list.push(root.val);
}
```

=== "TS"

```typescript title="binary_tree_dfs.ts"
/* 前序遍历 */
function preOrder(root: TreeNode | null): void {
    if (root === null) {
        return;
    }
    // 访问优先级：根节点 -> 左子树 -> 右子树
    list.push(root.val);
    preOrder(root.left);
    preOrder(root.right);
}

/* 中序遍历 */
function inOrder(root: TreeNode | null): void {
    if (root === null) {
        return;
    }
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root.left);
    list.push(root.val);
    inOrder(root.right);
}

/* 后序遍历 */
function postOrder(root: TreeNode | null): void {
    if (root === null) {
        return;
    }
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root.left);
    postOrder(root.right);
    list.push(root.val);
}
```

=== "Dart"

```dart title="binary_tree_dfs.dart"
/* 前序遍历 */
void preOrder(TreeNode? node) {
  if (node == null) return;
  // 访问优先级：根节点 -> 左子树 -> 右子树
  list.add(node.val);
  preOrder(node.left);
  preOrder(node.right);
}

/* 中序遍历 */
void inOrder(TreeNode? node) {
  if (node == null) return;
  // 访问优先级：左子树 -> 根节点 -> 右子树
  inOrder(node.left);
  list.add(node.val);
  inOrder(node.right);
}

/* 后序遍历 */
void postOrder(TreeNode? node) {
  if (node == null) return;
  // 访问优先级：左子树 -> 右子树 -> 根节点
  postOrder(node.left);
  postOrder(node.right);
  list.add(node.val);
}
```

=== "Rust"

```rust title="binary_tree_dfs.rs"
/* 前序遍历 */
fn pre_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
    let mut result = vec![];

    if let Some(node) = root {
        // 访问优先级：根节点 -> 左子树 -> 右子树
        result.push(node.borrow().val);
        result.extend(pre_order(node.borrow().left.as_ref()));
        result.extend(pre_order(node.borrow().right.as_ref()));
    }
    result
}

/* 中序遍历 */
fn in_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
    let mut result = vec![];

    if let Some(node) = root {
        // 访问优先级：左子树 -> 根节点 -> 右子树
        result.extend(in_order(node.borrow().left.as_ref()));
        result.push(node.borrow().val);
        result.extend(in_order(node.borrow().right.as_ref()));
    }
    result
}

/* 后序遍历 */
fn post_order(root: Option<&Rc<RefCell<TreeNode>>>) -> Vec<i32> {
    let mut result = vec![];

    if let Some(node) = root {
        // 访问优先级：左子树 -> 右子树 -> 根节点
        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"
/* 前序遍历 */
void preOrder(TreeNode *root, int *size) {
    if (root == NULL)
        return;
    // 访问优先级：根节点 -> 左子树 -> 右子树
    arr[(*size)++] = root->val;
    preOrder(root->left, size);
    preOrder(root->right, size);
}

/* 中序遍历 */
void inOrder(TreeNode *root, int *size) {
    if (root == NULL)
        return;
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root->left, size);
    arr[(*size)++] = root->val;
    inOrder(root->right, size);
}

/* 后序遍历 */
void postOrder(TreeNode *root, int *size) {
    if (root == NULL)
        return;
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root->left, size);
    postOrder(root->right, size);
    arr[(*size)++] = root->val;
}
```

=== "Kotlin"

```kotlin title="binary_tree_dfs.kt"
/* 前序遍历 */
fun preOrder(root: TreeNode?) {
    if (root == null) return
    // 访问优先级：根节点 -> 左子树 -> 右子树
    list.add(root._val)
    preOrder(root.left)
    preOrder(root.right)
}

/* 中序遍历 */
fun inOrder(root: TreeNode?) {
    if (root == null) return
    // 访问优先级：左子树 -> 根节点 -> 右子树
    inOrder(root.left)
    list.add(root._val)
    inOrder(root.right)
}

/* 后序遍历 */
fun postOrder(root: TreeNode?) {
    if (root == null) return
    // 访问优先级：左子树 -> 右子树 -> 根节点
    postOrder(root.left)
    postOrder(root.right)
    list.add(root._val)
}
```

=== "Ruby"

```ruby title="binary_tree_dfs.rb"
### 前序遍历 ###
def pre_order(root)
  return if root.nil?

  # 访问优先级：根节点 -> 左子树 -> 右子树
  $res << root.val
  pre_order(root.left)
  pre_order(root.right)
end

### 中序遍历 ###
def in_order(root)
  return if root.nil?

  # 访问优先级：左子树 -> 根节点 -> 右子树
  in_order(root.left)
  $res << root.val
  in_order(root.right)
end

### 后序遍历 ###
def post_order(root)
  return if root.nil?

  # 访问优先级：左子树 -> 右子树 -> 根节点
  post_order(root.left)
  post_order(root.right)
  $res << root.val
end
```

=== "Zig"

```zig title="binary_tree_dfs.zig"
// 前序遍历
fn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {
    if (root == null) return;
    // 访问优先级：根节点 -> 左子树 -> 右子树
    try list.append(root.?.val);
    try preOrder(T, root.?.left);
    try preOrder(T, root.?.right);
}

// 中序遍历
fn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {
    if (root == null) return;
    // 访问优先级：左子树 -> 根节点 -> 右子树
    try inOrder(T, root.?.left);
    try list.append(root.?.val);
    try inOrder(T, root.?.right);
}

// 后序遍历
fn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {
    if (root == null) return;
    // 访问优先级：左子树 -> 右子树 -> 根节点
    try postOrder(T, root.?.left);
    try postOrder(T, root.?.right);
    try list.append(root.?.val);
}
```

??? pythontutor "Code Visualization"

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src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%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%E8%8A%82%E7%82%B9%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%E8%8A%82%E7%82%B9%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%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%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%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%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%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%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%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%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%E9%81%8D%E5%8E%86%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%AE%BF%E9%97%AE%E4%BC%98%E5%85%88%E7%BA%A7%EF%BC%9A%E6%A0%B9%E8%8A%82%E7%82%B9%20-%3E%20%E5%B7%A6%E5%AD%90%E6%A0%91%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A0%91%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%E9%81%8D%E5%8E%86%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%AE%BF%E9%97%AE%E4%BC%98%E5%85%88%E7%BA%A7%EF%BC%9A%E5%B7%A6%E5%AD%90%E6%A0%91%20-%3E%20%E6%A0%B9%E8%8A%82%E7%82%B9%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A0%91%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%90%8E%E5%BA%8F%E9%81%8D%E5%8E%86%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%AE%BF%E9%97%AE%E4%BC%98%E5%85%88%E7%BA%A7%EF%BC%9A%E5%B7%A6%E5%AD%90%E6%A0%91%20-%3E%20%E5%8F%B3%E5%AD%90%E6%A0%91%20-%3E%20%E6%A0%B9%E8%8A%82%E7%82%B9%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%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E8%BF%99%E9%87%8C%E5%80%9F%E5%8A%A9%E4%BA%86%E4%B8%80%E4%B8%AA%E4%BB%8E%E6%95%B0%E7%BB%84%E7%9B%B4%E6%8E%A5%E7%94%9F%E6%88%90%E4%BA%8C%E5%8F%89%E6%A0%91%E7%9A%84%E5%87%BD%E6%95%B0%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1,%202,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20pre_order%28root%29%0A%20%20%20%20print%28%22%5Cn%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%E7%9A%84%E8%8A%82%E7%82%B9%E6%89%93%E5%8D%B0%E5%BA%8F%E5%88%97%20%3D%20%22,%20res%29%0A%0A%20%20%20%20%23%20%E4%B8%AD%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20res.clear%28%29%0A%20%20%20%20in_order%28root%29%0A%20%20%20%20print%28%22%5Cn%E4%B8%AD%E5%BA%8F%E9%81%8D%E5%8E%86%E7%9A%84%E8%8A%82%E7%82%B9%E6%89%93%E5%8D%B0%E5%BA%8F%E5%88%97%20%3D%20%22,%20res%29%0A%0A%20%20%20%20%23%20%E5%90%8E%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20res.clear%28%29%0A%20%20%20%20post_order%28root%29%0A%20%20%20%20print%28%22%5Cn%E5%90%8E%E5%BA%8F%E9%81%8D%E5%8E%86%E7%9A%84%E8%8A%82%E7%82%B9%E6%89%93%E5%8D%B0%E5%BA%8F%E5%88%97%20%3D%20%22,%20res%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=129&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> 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!!! tip

Depth-first search can also be implemented based on iteration, interested readers can study this on their own.

The 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".

"Recursion" means starting a new method, the program accesses the next node in this process.
"Return" means the function returns, indicating the current node has been fully accessed.

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Figure 7-11 The recursive process of preorder traversal

2. Complexity analysis

Time complexity is O(n): All nodes are visited once, using O(n) time.
Space complexity is O(n): In the worst case, i.e., the tree degrades into a linked list, the recursion depth reaches n, the system occupies O(n) stack frame space.

46 KiB Raw Blame History

7.2 Binary tree traversal

7.2.1 Level-order traversal

1. Code implementation

2. Complexity analysis

7.2.2 Preorder, inorder, and postorder traversal

1. Code implementation

2. Complexity analysis

46 KiB

Raw Blame History