Vertical Order Traversal of a Binary Tree

To calculate the vertical order traversal of a binary tree, we can perform a depth-first search (DFS) and keep track of the column and row numbers for each node. We’ll then sort the nodes by their column numbers and row numbers. If two nodes are in the same row and column, we’ll sort them by their values.

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from typing import List, Optional
from collections import defaultdict

# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, val=0, left=None, right=None):
#         self.val = val
#         self.left = left
#         self.right = right

class Solution:
    def verticalTraversal(self, root: Optional[TreeNode]) -> List[List[int]]:
        # Dictionary to hold nodes in corresponding columns
        column_map = defaultdict(list)

        def dfs(node, row, col):
            if not node:
                return
            # Add the node value to the corresponding column
            column_map[col].append((row, node.val))
            # Explore left and right children
            dfs(node.left, row + 1, col - 1)
            dfs(node.right, row + 1, col + 1)

        # Start DFS from the root at position (0, 0)
        dfs(root, 0, 0)

        # Sort columns and build the result
        result = []
        for col in sorted(column_map.keys()):
            # Sort nodes in the same column by row number, then by value
            nodes = sorted(column_map[col])
            result.append([value for _, value in nodes])

        return result

The dfs function performs a depth-first search of the tree, keeping track of the row and column for each node. The column_map dictionary is used to store the nodes in their corresponding columns.

Finally, the result is constructed by sorting the columns and then sorting the nodes in each column by their row number and value.

This code returns the vertical order traversal of the given binary tree, satisfying the problem’s requirements.

Identifying Problem Isomorphism

“Vertical Order Traversal of a Binary Tree” requires us to return the vertical order traversal of its nodes’ values. It involves traversing the tree and mapping nodes to their vertical order, followed by sorting and grouping.

An isomorphic problem to this is “Binary Tree Level Order Traversal” (LeetCode #102). In this problem, we’re asked to traverse a binary tree level by level from left to right.

Both involve traversing a binary tree and aggregating node values in a specific order. “Vertical Order Traversal of a Binary Tree” has an additional dimension of sorting by vertical order and within the same vertical order, sorting by level, which is not required in the “Binary Tree Level Order Traversal” problem. Thus, this isomorphism is approximate as both involve tree traversal and value aggregation, but the order of aggregation is different.

10 Prerequisite LeetCode Problems

This requires understanding of binary tree traversal, sorting, and hashing. Here are 10 problems as preparation:

“94. Binary Tree Inorder Traversal”: This is a basic problem to understand the concept of tree traversal, which is necessary to solve the main problem.
“102. Binary Tree Level Order Traversal”: This problem is about traversing a binary tree level by level, which is related to the vertical order traversal.
“107. Binary Tree Level Order Traversal II”: This problem is a little bit more complicated than the previous one because the traversal is bottom-up instead of top-down.
“144. Binary Tree Preorder Traversal”: This is another tree traversal problem, but this time it’s about preorder traversal.
“145. Binary Tree Postorder Traversal”: And here is the postorder traversal problem.
“429. N-ary Tree Level Order Traversal”: In this problem, you will extend your knowledge of level order traversal from binary trees to N-ary trees.
“103. Binary Tree Zigzag Level Order Traversal”: This problem requires zigzag level order traversal, which can help you to understand more complex traversal patterns.
“314. Binary Tree Vertical Order Traversal”: This problem is a direct precursor to the given problem. It requires vertical order traversal, but without considering the second sorting condition.
“105. Construct Binary Tree from Preorder and Inorder Traversal”: This problem can help you understand more about the structure of binary trees.
“106. Construct Binary Tree from Inorder and Postorder Traversal”: And this problem is a good follow-up to the previous one.

Problem Classification

Problem Statement:Given the root of a binary tree, calculate the vertical order traversal of the binary tree.

For each node at position (row, col), its left and right children will be at positions (row + 1, col - 1) and (row + 1, col + 1) respectively. The root of the tree is at (0, 0).

The vertical order traversal of a binary tree is a list of top-to-bottom orderings for each column index starting from the leftmost column and ending on the rightmost column. There may be multiple nodes in the same row and same column. In such a case, sort these nodes by their values.

Return the vertical order traversal of the binary tree.

Example 1:

Input: root = [3,9,20,null,null,15,7] Output: [[9],[3,15],[20],[7]] Explanation: Column -1: Only node 9 is in this column. Column 0: Nodes 3 and 15 are in this column in that order from top to bottom. Column 1: Only node 20 is in this column. Column 2: Only node 7 is in this column.

Example 2:

Input: root = [1,2,3,4,5,6,7] Output: [[4],[2],[1,5,6],[3],[7]] Explanation: Column -2: Only node 4 is in this column. Column -1: Only node 2 is in this column. Column 0: Nodes 1, 5, and 6 are in this column. 1 is at the top, so it comes first. 5 and 6 are at the same position (2, 0), so we order them by their value, 5 before 6. Column 1: Only node 3 is in this column. Column 2: Only node 7 is in this column.

Example 3:

Input: root = [1,2,3,4,6,5,7] Output: [[4],[2],[1,5,6],[3],[7]] Explanation: This case is the exact same as example 2, but with nodes 5 and 6 swapped. Note that the solution remains the same since 5 and 6 are in the same location and should be ordered by their values.

Constraints:

The number of nodes in the tree is in the range [1, 1000]. 0 <= Node.val <= 1000

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.

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.

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?

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

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

Thought Process

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.

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?

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.