Algorithms 101: PostOrder Traversal

后序遍历

PostOrder traversal is a tree traversal method where the root node is visited after its children. It is one of the depth-first search (DFS) methods used to explore all the nodes in a tree. In PostOrder traversal, the process is to recursively visit the left subtree first, then the right subtree, and finally, visit the root node. This method ensures that the root node is always visited last, after all its descendants have been visited.
后序遍历是一种树遍历方法，其中根节点在其子节点之后被访问。它是深度优先搜索 (DFS) 方法之一，用于遍历树中的所有节点。在后序遍历中，过程是首先递归访问左子树，然后右子树，最后访问根节点。这种方法确保在所有子节点被访问后，根节点总是最后被访问。

How PostOrder Traversal Works 后序遍历如何工作

Recursively traverse the left subtree in PostOrder.
以后序递归遍历左子树。
Recursively traverse the right subtree in PostOrder.
以后序递归遍历右子树。
Visit the root node.
访问根节点。

This method ensures that the root node is visited last, making it useful for operations where children need to be processed before their parent node.
这种方法确保根节点最后被访问，这对于需要在处理父节点之前处理其子节点的操作非常有用。

Example of PostOrder Traversal 后序遍历的示例

Consider the following binary tree:
考虑以下二叉树：

Step-by-Step Process of PostOrder Traversal 后序遍历的分步过程

Start with the Left Subtree:
从左子树开始：
- Move to the leftmost node, starting with node 2.
  移动到最左边的节点，从节点 2 开始。
- Move to node 4 (left child of 2).
  移动到节点 4（2 的左子节点）。
- Visit node 4 after traversing its (non-existent) subtrees.
  在遍历其（不存在的）子树后访问节点 4。
Backtrack to Root of Left Subtree:
回溯到左子树的根节点：
- Move back to node 2, then move to node 5 (right child of 2).
  回到节点 2，然后移动到节点 5（2 的右子节点）。
- Visit node 5.
  访问节点 5。
Visit the Root of Left Subtree:
访问左子树的根节点：
- After visiting both children of node 2, visit node 2.
  在访问完节点 2 的两个子节点后，访问节点 2。
Traverse the Right Subtree:
遍历右子树：
- Move to node 3 (right child of 1).
  移动到节点 3（1 的右子节点）。
- Move to node 6 (left child of 3).
  移动到节点 6（3 的左子节点）。
- Visit node 6 after traversing its (non-existent) subtrees.
  在遍历其（不存在的）子树后访问节点 6。
Backtrack to Root of Right Subtree:
回溯到右子树的根节点：
- Move back to node 3, then move to node 7 (right child of 3).
  回到节点 3，然后移动到节点 7（3 的右子节点）。
- Visit node 7.
  访问节点 7。
Visit the Root Node:
访问根节点：
- After visiting both children of node 3, visit node 3.
  在访问完节点 3 的两个子节点后，访问节点 3。
- Finally, visit the root node 1 after all its children have been visited.
  最后，在访问完所有子节点后访问根节点 1。

Final PostOrder Traversal: [4, 5, 2, 6, 7, 3, 1]
最终后序遍历：[4, 5, 2, 6, 7, 3, 1]

Python Implementation of PostOrder Traversal 后序遍历的Python实现

Here is the Python code to implement PostOrder traversal of a binary tree:
以下是实现二叉树后序遍历的Python代码：

class Node:
    def __init__(self, key):
        self.left = None
        self.right = None
        self.val = key

def postorder_traversal(root):
    if root:
        postorder_traversal(root.left)
        postorder_traversal(root.right)
        print(root.val, end=' ')

# Creating the binary tree from the example
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
root.right.left = Node(6)
root.right.right = Node(7)

postorder_traversal(root)

This code will output:
这段代码的输出将是：

4 5 2 6 7 3 1

Analysis of PostOrder Traversal 后序遍历的分析

Time Complexity: O(n) because each node is visited exactly once.
时间复杂度：O(n)，因为每个节点都被访问一次。
Space Complexity: O(h) where h is the height of the tree, due to the recursive call stack.
空间复杂度：O(h)，其中 h 是树的高度，因为递归调用栈。
Use Cases: PostOrder traversal is useful in scenarios where you need to delete or process child nodes before the parent node, such as deleting a tree, evaluating an expression tree, or executing a task hierarchy.
使用案例：后序遍历在需要在父节点之前删除或处理子节点的场景中特别有用，例如删除树、评估表达式树或执行任务层次结构。

Example with a Different Tree 示例：使用不同的树

Consider the following binary tree:
考虑以下二叉树：

    A
   / \
  B   C
 / \
D   E

Performing a PostOrder traversal on this tree will yield:
对这棵树执行后序遍历将产生：

D E B C A

Conclusion 结论

PostOrder traversal is a systematic tree traversal method where the root node is visited after its children. It is particularly useful in applications where you need to process or delete child nodes before their parent node. This traversal method is commonly used in algorithms that need to evaluate or dismantle a tree structure from the leaves up to the root.
后序遍历是一种系统的树遍历方法，其中根节点在其子节点之后被访问。它在需要在父节点之前处理或删除子节点的应用中特别有用。此遍历方法常用于从叶子到根节点评估或拆除树结构的算法中。

This method is easy to implement recursively and is an essential tool in tree-related algorithms, especially in tasks that involve hierarchical processing.
这种方法很容易递归实现，并且是树相关算法中的一种基本工具，特别是在涉及层次处理的任务中。

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