Encode N-ary Tree to Binary Tree

To encode an N-ary tree into a binary tree and decode the binary tree to get the original N-ary tree, we can utilize the left child of a binary tree to represent the first child in the N-ary tree, and the right child of the binary tree to represent the sibling of the N-ary tree.

Here’s how we can design the encoding and decoding process:

Encoding

The left child of a binary tree node represents the first child of the corresponding N-ary tree node.
The right child of a binary tree node represents the next sibling of the corresponding N-ary tree node.
Iterate through the children of the N-ary tree node and recursively build the binary tree using the above rules.

Decoding

The decoding process is a reverse operation of encoding.
Reconstruct the N-ary tree by recognizing that the left child of a binary tree node corresponds to the first child of the N-ary tree node, and the right child of the binary tree node corresponds to the sibling of the N-ary tree node.

Code:

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class Codec:
    # Encodes an n-ary tree to a binary tree.
    def encode(self, root: 'Optional[Node]') -> Optional[TreeNode]:
        if not root:
            return None

        binary_tree_root = TreeNode(root.val)
        if root.children:
            binary_tree_root.left = self.encode(root.children[0])

        curr = binary_tree_root.left
        for child in root.children[1:]:
            curr.right = self.encode(child)
            curr = curr.right

        return binary_tree_root

    # Decodes your binary tree to an n-ary tree.
    def decode(self, data: Optional[TreeNode]) -> 'Optional[Node]':
        if not data:
            return None

        n_ary_tree_root = Node(data.val, [])
        curr = data.left
        while curr:
            n_ary_tree_root.children.append(self.decode(curr))
            curr = curr.right

        return n_ary_tree_root

This algorithm ensures that an N-ary tree can be encoded into a binary tree and that this binary tree can be decoded into the original N-nary tree structure. The time and space complexity of both the encode and decode functions are O(N), where N is the number of nodes in the tree.

Identifying Problem Isomorphism

“Encode N-ary Tree to Binary Tree” can be mapped to “Flatten Binary Tree to Linked List”.

The crux of both problems involves transforming one type of tree structure into another. In “Encode N-ary Tree to Binary Tree”, you have to convert an N-ary tree (a tree where each node can have N number of children) into a binary tree. On the other hand, “Flatten Binary Tree to Linked List” requires you to flatten a binary tree (where each node can have at most two children) into a linked list.

Both tasks involve restructuring tree nodes while preserving certain relationships. The solutions to these problems require a good understanding of tree traversal techniques and tree properties.

“Flatten Binary Tree to Linked List” is simpler as it reduces the structure complexity from a binary tree to a linear structure (linked list), whereas “Encode N-ary Tree to Binary Tree” involves a more complex transformation with more child nodes to consider.

10 Prerequisite LeetCode Problems

“431. Encode N-ary Tree to Binary Tree” involves tree data structures and specifically converting an N-ary tree to a binary tree. Here are some simpler problems to prepare for this:

LeetCode 104. Maximum Depth of Binary Tree
- This problem introduces the concept of traversing a binary tree and calculating its depth.
LeetCode 116. Populating Next Right Pointers in Each Node
- This problem involves dealing with a perfect binary tree and understanding the structure of a binary tree.
LeetCode 429. N-ary Tree Level Order Traversal
- This problem will help you understand how to traverse an N-ary tree.
LeetCode 102. Binary Tree Level Order Traversal
- This problem will help you understand how to traverse a binary tree, a concept that is crucial for the “Encode N-ary Tree to Binary Tree” problem.
LeetCode 589. N-ary Tree Preorder Traversal
- This problem introduces the concept of preorder traversal for an N-ary tree.
LeetCode 297. Serialize and Deserialize Binary Tree
- This problem introduces the concept of encoding and decoding a binary tree, which is similar to the “Encode N-ary Tree to Binary Tree” problem.
LeetCode 133. Clone Graph
- This problem is about deep copying a graph, a concept that can help you understand how to copy and transform a tree.
LeetCode 114. Flatten Binary Tree to Linked List
- This problem requires you to modify the structure of a binary tree, which is similar to the task of encoding an N-ary tree into a binary tree.
LeetCode 100. Same Tree
- This problem involves comparing two trees, which can help you understand how to construct and validate a binary tree from an N-ary tree.
LeetCode 559. Maximum Depth of N-ary Tree
- This problem helps you understand the structure and depth of an N-ary tree, a concept that’s used in the “Encode N-ary Tree to Binary Tree” problem.

By practicing these problems, you’ll gain a solid understanding of how to work with binary trees and N-ary trees, which are key to solving the “431. Encode N-ary Tree to Binary Tree” problem.

Problem Boundary

How to establish the boundary of this problem?

Problem Classification

Problem Statement:Design an algorithm to encode an N-ary tree into a binary tree and decode the binary tree to get the original N-ary tree. An N-ary tree is a rooted tree in which each node has no more than N children. Similarly, a binary tree is a rooted tree in which each node has no more than 2 children. There is no restriction on how your encode/decode algorithm should work. You just need to ensure that an N-ary tree can be encoded to a binary tree and this binary tree can be decoded to the original N-nary tree structure.

Nary-Tree input serialization is represented in their level order traversal, each group of children is separated by the null value (See following example).

For example, you may encode the following 3-ary tree to a binary tree in this way:

Input: root = [1,null,3,2,4,null,5,6] Note that the above is just an example which might or might not work. You do not necessarily need to follow this format, so please be creative and come up with different approaches yourself.

Example 1:

Input: root = [1,null,3,2,4,null,5,6] Output: [1,null,3,2,4,null,5,6]

Example 2:

Input: root = [1,null,2,3,4,5,null,null,6,7,null,8,null,9,10,null,null,11,null,12,null,13,null,null,14] Output: [1,null,2,3,4,5,null,null,6,7,null,8,null,9,10,null,null,11,null,12,null,13,null,null,14] Example 3:

Input: root = [] Output: []

Constraints:

The number of nodes in the tree is in the range [0, 104]. 0 <= Node.val <= 104 The height of the n-ary tree is less than or equal to 1000 Do not use class member/global/static variables to store states. Your encode and decode algorithms should be stateless.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

In order to have me distill a problem to its core, you could ask questions that prompt for a deeper analysis of the problem, understanding of the underlying concepts, and simplification of the problem’s essence. Here are some examples of such prompts:

Can you identify the fundamental concept or principle this problem is based upon? Please explain.
What is the simplest way you would describe this problem to someone unfamiliar with the subject?
What is the core problem we are trying to solve? Can we simplify the problem statement?
Can you break down the problem into its key components?
What is the minimal set of operations we need to perform to solve this problem?

These prompts guide the discussion towards simplifying the problem, stripping it down to its essential elements, and understanding the core problem to be solved.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

To explain a problem in simple, non-technical language:

Can you explain [problem] in simple terms or like you would explain to a non-technical person?
Imagine you’re explaining [problem] to someone without a background in programming. How would you describe it?
If you had to explain [problem] to a child or someone who doesn’t know anything about coding, how would you do it?
In layman’s terms, how would you explain the concept of [problem]?
Could you provide a metaphor or everyday example to explain the idea of [problem]?

These prompts encourage an explanation that avoids jargon or assumes minimal technical knowledge, making the concept more accessible to a broader audience.

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

Problem Name:
- What is the problem that you are trying to solve?
Method Name:
- What is the name of the method/function that you are using to solve this problem?
Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

By answering these questions for each method in your program, you can ensure that you have a clear understanding of what each part of your code is doing and how it should behave. This will help prevent bugs and make your code easier to read and maintain.

Problem Decomposition

Problem Name:
- What is the complex problem that you are trying to solve?
Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

By going through these steps for each complex problem, you can break it down into manageable parts, making it much easier to devise an effective solution.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

What are the reasons for making these mistakes in the given code?