As part of a research task, I came across a sub-problem as follows:

Given an undirected, unweighted tree of $$$n$$$ nodes and a threshold $$$k$$$, partition the $$$n-1$$$ edges of the tree into simple paths such that every edge belongs to exactly one simple path and the maximum length of a simple path is atmost $$$k$$$. What is the minimum number of simple path partitions which meet this criteria? Note that every edge belongs to exactly one simple path, but the same vertices can be visited by multiple simple paths.

I was wondering if this is by any chance a "standard" problem with a known polynomial time solution?

In a recent exchange, kHarsh3715 and I were discussing the potential topics in Number Theory to learn. kHarsh3715, with conviction, deemed Mobius Inversion, Convolutions, and Diophantine Equations as "very useful topics." He further went on to include Generating Functions, Linear Recurrences, and ventured into the arcane with the Lucas Theorem, Pollard Rho, and Primality Testing.

Baffled by his bold assertions, I simply replied, "Umnik would disagree" (Referring to this blog of course: link). This simple statement sparked a fierce debate, escalating to a wager that put our very financial futures at stake. Only the legendary Um_nik himself could resolve this epic conflict and determine who would emerge victorious.