  In a previous paper, we introduced the Collatz polynomials $P_N(z)$, whose
coefficients are the terms of the Collatz sequence of the positive integer $N$.
Our work in this paper expands on our previous results, using the
Enestr\"om-Kakeya Theorem to tighten our old bounds of the roots of $P_N(z)$
and giving precise conditions under which these new bounds are sharp. In
particular, we confirm an experimental result that zeros on the circle
$\{z\in\mathbb{C}: |z| = 2\}$ are rare: the set of $N$ such that $P_N(z)$ has a
root of modulus 2 is sparse in the natural numbers. We close with some
questions for further study.
