Problem Overview

Given a k-th order Fibonacci tree represented by its pre-order traversal of node values, find the path between two given nodes. The challenge is to solve this without explicitly building the tree, using mathematical properties of Fibonacci trees to calculate node positions directly.

This problem tests your understanding of:

• Fibonacci tree structure and mathematical properties

• Tree traversal algorithms and their time complexity

• Optimization techniques to avoid unnecessary tree construction

• Lowest Common Ancestor (LCA) algorithms

Fibonacci Tree Definition

A k-th order Fibonacci tree is defined recursively:

• Order 0: Empty tree (0 nodes)

• Order 1: Single node (1 node)

• Order k (k >= 2): A tree with:

• Root node

• Left subtree: (k-1)-th order Fibonacci tree

• Right subtree: (k-2)-th order Fibonacci tree

The number of nodes follows: T(k) = 1 + T(k-1) + T(k-2) with T(0)=0, T(1)=1.

This gives: T(1)=1, T(2)=2, T(3)=4, , , ...

Formula: A k-th order Fibonacci tree has exactly F(k+2) - 1 nodes, where F is the standard Fibonacci sequence (F(0)=0, F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5...).

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Example Tree (k=4):

Pre-order values: 1, 2, 3, 4, 5, 6, 7

Tree structure (T(4) = 7 nodes):

1

/ \

2 6

/ \ /

3 5 7

/

4

Left subtree of root (1): T(3) = 4 nodes (values 2-5)

Right subtree of root (1): T(2) = 2 nodes (values 6-7)

Find path from node 4 to node 7:

result = get_path(k=4, start=4, end=7)

Output: [4, 3, 2, 1, 6, 7]

Explanation: Path goes up to LCA (node 1), then down to target

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• 1 <= k <= 45 (tree order)

• Maximum nodes: T(45) = F(47) - 1 (approx 2.97 billion nodes)

• 1 <= start, end <= T(k) (valid node values in pre-order traversal)

• Tree uses 1-based indexing for node values

• All node values in the tree are unique