N-ary Tree Postorder Traversal

Postorder traversal is a common tree traversal technique that visits all the nodes of a tree in a specific order: left children, then right children, and finally the node itself. In the case of an n-ary tree, where a node can have more than two children, the postorder traversal will visit all the children before visiting the node itself.

The following is an approach for postorder traversal of an n-ary tree:

Recursive Approach

  1. Initialize the Result: Start with an empty list to store the result of the traversal.
  2. Recursion Function: Create a recursive function that takes a node and the result list.
    • Base Case: If the node is None, simply return.
    • Recursive Case: For each child of the current node, call the recursion function.
    • Add to Result: Append the value of the current node to the result list.
  3. Call the Function: Call the recursive function starting with the root of the tree.
  4. Return the Result: Return the result list.

Code

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class Solution:
    def postorder(self, root: 'Node') -> List[int]:
        result = []

        def traverse(node):
            if not node:
                return
            for child in node.children:
                traverse(child)
            result.append(node.val)

        traverse(root)
        return result

Explanation

  • The function traverse is defined to handle the recursive traversal of the tree.
  • For each node, it iterates through its children and recursively calls traverse on them.
  • Once all children have been visited, it appends the current node’s value to the result list.

Insights

The time complexity is (O(n)), where (n) is the number of nodes in the tree. Every node is visited exactly once. The space complexity is also (O(n)) for the recursion call stack, considering the worst case where the tree is highly unbalanced.

This code relies on recursive nature of tree structures.

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     def iterative(self, root):
        if not root: return []
        stack = [root]
        out = []
        while len(stack):
            top = stack.pop()
            out.append(top.val)
            stack.extend(top.children or [])
        return out[::-1]

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# Definition for a Node.
# class Node
#     attr_accessor :val, :children
#     def initialize(val)
#         @val = val
#         @children = []
#     end
# end

# @param {Node} root
# @return {List[int]}
def postorder(root)
    output = []
    stack = []

    if root.nil?
        return output
    end

    stack.push(root)

    while !stack.empty?
       node = stack.pop

       output << node.val

        for c in node.children
            stack.push(c) 
        end

    end
    return output.reverse
end

Problem Classification

This problem belongs to the domain of data structures, specifically trees. More precisely, it’s about the traversal of an n-ary tree.

  1. ‘What’ Components:

    • Input: The root of an n-ary tree.
    • Output: The postorder traversal of the tree’s nodes’ values.
    • Unspecified details: The problem doesn’t specify the exact type or size of the values stored in the tree nodes, implying that the solution should be able to handle any reasonable input.

This is a tree traversal problem, requiring an understanding of postorder traversal. The problem can be classified as a traversal or search problem. These types of problems require navigating the tree in a specific order (in this case, postorder: left -> right -> root) and performing an operation at each node (in this case, recording the node’s value).

The problem can also be classified as a depth-first search (DFS) problem because postorder traversal is a type of DFS where the root is processed last. DFS problems involve exploring as far as possible along each branch before backtracking.

Understanding how to construct and manipulate tree data structures is crucial for solving this problem. Specifically, navigate the tree correctly and store the values in the desired order.

Language Agnostic Coding Drills

  1. Dissection of Code into Concepts:

    • Data structure initialization: The code initiates several data structures (a stack and a list).

    • Null check: The first condition checks if the root of the tree is null or not.

    • Iteration over a data structure: The ‘while’ loop iterates over the stack until it is empty.

    • Data structure manipulation (Stack): The ‘pop’ method is used to remove the top element from the stack.

    • Accessing attributes of an object: Accessing the ‘val’ and ‘children’ attributes of the node objects.

    • Appending to a list: The ‘append’ and ’extend’ methods are used to add elements to the end of a list.

    • List reversal: The list slicing technique is used to reverse the list of node values.

  2. Order of Coding Concepts (increasing difficulty):

    • Data structure initialization: Basic understanding of how to declare variables and data structures.

    • Appending to a list: Understanding of how to add elements to a list in Python.

    • Null check: Basic understanding of how to perform null or none check in Python.

    • Accessing attributes of an object: Understanding of object-oriented programming concepts and how to access attributes of objects in Python.

    • Iteration over a data structure: Understand how to iterate over a data structure using a while loop, which requires an understanding of loop control flow.

    • Data structure manipulation (Stack): Deeper understanding of the stack data structure and its LIFO (Last In First Out) property.

    • List reversal: Requires knowledge of list slicing technique and its parameters for reversing a list in Python.

  3. Problem-Solving Approach:

    • Understanding the problem and identifying requirements: Identify that the problem is about post-order traversal of an n-ary tree, which involves visiting each node’s children before the node itself.

    • Mapping problem to relevant concepts: Identify the main coding concepts relevant to this problem, such as tree traversal, stack manipulation, list operations, and recursion.

    • Building the solution step-by-step: Begin by initializing the necessary data structures and conducting a null check on the root. Implement the core tree traversal logic, which involves iterating over the tree nodes, visiting the child nodes before the parent node, and maintaining a stack to track nodes to visit. Append each node’s value to a list as it is visited.

    • Integration and final touch: After visiting all nodes, reverse the list of node values to get the postorder traversal, as the stack-based method essentially results in a reverse postorder traversal.

    • Testing and validation: Test the solution with various cases, including edge cases, to ensure it meets the problem requirements and handles potential edge cases correctly.

Targeted Drills in Python

  1. Python Code for each Identified Concept
  • Data structure initialization
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list_example = []   # Initializing a list
stack_example = []  # Initializing a stack as a list
  • Appending to a list
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list_example.append("item")  # Appending an item to a list
  • Null check
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if variable is not None:  # Check if a variable is not None
    print("Variable is not None")
  • Accessing attributes of an object
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class Node:   # Define a class Node
    def __init__(self, value):
        self.value = value  # Initialize an attribute `value`

node_example = Node(5)  # Create an object of class Node
print(node_example.value)  # Accessing the attribute `value` of the object
  • Iteration over a data structure
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stack_example = [1, 2, 3, 4, 5]  # Initializing a stack as a list
while stack_example:  # Iteration over stack until it's empty
    top = stack_example.pop()  # pop top element from stack
    print(top)
  • Data structure manipulation (Stack)
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stack_example = [1, 2, 3, 4, 5]  # Initializing a stack as a list
top_element = stack_example.pop()  # Pop top element from stack
print(top_element)
  • List reversal
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list_example = [1, 2, 3, 4, 5]  # Initializing a list
list_example_reversed = list_example[::-1]  # Reversing a list
print(list_example_reversed)
  1. Problem-Specific Concept
  • Postorder traversal of an n-ary tree

This problem is specific to postorder traversal in an n-ary tree. Postorder traversal of a tree involves visiting the children of a node before the node itself. For an n-ary tree, this means visiting all of the node’s children from left to right, and then visiting the node itself.

  1. Integration of Drills
  • Initialize the necessary data structures (a stack and a list).
  • Check if the root node is None. If it is, return an empty list.
  • Add the root node to the stack.
  • Start a loop that continues until the stack is empty. In each iteration:
    • Pop the top node from the stack.
    • Append the value of the node to the list.
    • Add the children of the node to the stack.
  • After the loop finishes (the stack is empty), reverse the list and return it.

The code can be integrated as follows:

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def postorder_traversal(root):
    if root is None:
        return []
    stack = [root]
    output = []
    while stack:
        node = stack.pop()
        output.append(node.val)
        for child in node.children:
            stack.append(child)
    return output[::-1]

Remember, this function is expected to work with an n-ary tree data structure, where each node has a value and a list of its children.