2-adic Valuations of Quadratic Sequences

Properties of 2-adic valuation sequences for generic quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at terminating nodes. As a corollary, we characterize zeros of generic quadratic polynomials with respect to membership in the field of 2-adics and the ring of 2-adic integers. Throughout, we frame results in terms of trees and sequences.