  Let $K$ be a complete, algebraically closed non-archimedean valued field, and
let $\varphi(z) \in K(z)$ have degree two. We describe the crucial set of
$\varphi$ in terms of the multipliers of $\varphi$ at the classical fixed
points, and use this to show that the crucial set determines a stratification
of the moduli space $\mathcal{M}_2(K)$ related to the reduction type of
$\varphi$. We apply this to settle a special case of a conjecture of Hsia
regarding the density of repelling periodic points in the non-archimedean Julia
set.
