Products of p-Adic Valuation Trees

The study of prime divisibility plays a crucial role in number theory. The p-adic valuation of a number is the highest power of a prime, p, that divides that number. Using this valuation, we construct p-adic valuation trees to visually represent the valuations of a sequence. We investigate how nodes split on trees generated by linear functions with rational coefficients, as well as those formed from a product of linear or lower degree polynomials. We describe the infinite branches of these polynomial trees and the valuations of their terminating nodes.