#1104

Medium

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Path In Zigzag Labelled Binary Tree

The problem involves finding the path from the root to a node in a zigzag-labelled binary tree.

Problem Statement

In an infinite binary tree, nodes are labelled in a specific pattern where each node has two children. The labelling occurs in row order, with odd-numbered rows (1st, 3rd, etc.) labelled from left to right, while even-numbered rows (2nd, 4th, etc.) are labelled right to left.

Given a node label, the task is to return the path from the root node (label 1) to the node with the given label. The key challenge lies in understanding the zigzag nature of the labelling and efficiently tracing the path from root to the target node.

Examples

Example 1

Input: label = 14

Output: [1,3,4,14]

Example details omitted.

Example 2

Input: label = 26

Output: [1,2,6,10,26]

Example details omitted.

Constraints

1 <= label <= 10^6

Solution Approach

Binary-tree traversal with state tracking

To solve the problem, the key approach involves simulating a binary-tree traversal while tracking the current row direction (left to right or right to left). The path is constructed by determining the parent node based on the zigzag pattern and adjusting the traversal accordingly.

Reversing direction for even rows

For even-numbered rows, the traversal direction must be reversed. This requires adjusting the way we compute parent-child relationships, reversing the label arithmetic to reflect the opposite direction of labelling.

Efficient path construction

By iteratively determining the parent of each node and constructing the path from the target node to the root, the solution can be achieved efficiently without the need to build the entire tree.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexities of the solution depend on the final approach used, particularly the method for determining parent nodes and constructing the path. Generally, the problem can be solved in logarithmic time relative to the label's size, with space complexity determined by the number of nodes in the path from the target node to the root.

What Interviewers Usually Probe

Look for familiarity with binary-tree traversal concepts and efficient path construction.
Evaluate how well the candidate handles the zigzag labelling pattern and row direction reversal.
Assess the candidate's ability to optimize the solution with respect to time and space complexity.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the zigzag labelling pattern, especially for even-numbered rows.
Incorrectly calculating the parent node, leading to errors in the path construction.
Not accounting for the fact that the problem asks for the path from the root to the given node, not the tree structure itself.

Follow-up variants

Determine the path for multiple target labels simultaneously.
Calculate the path for nodes in a binary tree with arbitrary labelling patterns.
Return the path for a node using a breadth-first search approach.

FAQ

What is the key concept for solving the Path In Zigzag Labelled Binary Tree problem?

The key concept is binary-tree traversal with state tracking, where the traversal direction alternates for each row, requiring careful computation of parent nodes.

How do you handle the zigzag nature of the labelling?

You handle the zigzag by adjusting the direction of traversal for even-numbered rows, reversing the label arithmetic to calculate the parent node.

What is the time complexity for solving this problem?

The time complexity is logarithmic relative to the label, as you only need to trace the path from the target node to the root.

Can this problem be solved using a breadth-first search?

While breadth-first search could be applied, an optimal solution uses direct traversal and path construction without needing a full search.

How does GhostInterview help with this problem?

GhostInterview guides you through the traversal process, helping you track the state of traversal and avoid common pitfalls in the zigzag labelling.

terminal

Solution

Solution 1: Mathematics

For a complete binary tree, the number of nodes in the $i$th row is $2^{i-1}$, and the range of node labels in the $i$th row is $[2^{i-1}, 2^i - 1]$. In the problem, for odd-numbered rows, the nodes are labeled from left to right, while for even-numbered rows, the nodes are labeled from right to left. Therefore, for the node $label$ in the $i$th row, its complementary node label is $2^{i-1} + 2^i - 1 - label$. So the actual parent node label of node $label$ is $(2^{i-1} + 2^i - 1 - label) / 2$. We can find the path from the root node to node $label$ by continuously finding the complementary node label and the parent node label until we reach the root node.

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class Solution:
    def pathInZigZagTree(self, label: int) -> List[int]:
        x = i = 1
        while (x << 1) <= label:
            x <<= 1
            i += 1
        ans = [0] * i
        while i:
            ans[i - 1] = label
            label = ((1 << (i - 1)) + (1 << i) - 1 - label) >> 1
            i -= 1
        return ans

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