  Given a number field $K$ and a polynomial $f(z) \in K[z]$, one can naturally
construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational
preperiodic points of $f$, with an edge $\alpha \to \beta$ if and only if
$f(\alpha) = \beta$. The dynamical uniform boundedness conjecture of Morton and
Silverman suggests that, for fixed integers $n \ge 1$ and $d \ge 2$, there are
only finitely many isomorphism classes of directed graphs $G(f,K)$ as one
ranges over all number fields $K$ of degree $n$ and polynomials $f(z) \in K[z]$
of degree $d$. In the case $(n,d) = (1,2)$, Poonen has given a complete
classification of all directed graphs which may be realized as
$G(f,\mathbb{Q})$ for some quadratic polynomial $f(z) \in \mathbb{Q}[z]$, under
the assumption that $f$ does not admit rational points of large period. The
purpose of the present article is to continue the work begun by the author,
Faber, and Krumm on the case $(n,d) = (2,2)$. By combining the results of the
previous article with a number of new results, we arrive at a partial result
toward a theorem like Poonen's -- with a similar assumption on points of large
period -- but over all quadratic extensions of $\mathbb{Q}$.
