The $p$-Adic Valuation Trees for Quadratic Polynomials for Odd Primes

We examine the behavior of the sequences of $p$-adic valuations of quadratic polynomials with integer coefficients for an odd prime $p$ through tree representations. Under this representation, a finite tree corresponds to a periodic sequence, and an infinite tree corresponds to an unbounded sequence. We use the polynomial coefficients to determine whether the $p$-adic valuation trees are finite or infinite, the number of infinite branches, the number of levels, the valuations at terminating nodes, and their relationship to the corresponding sequences.