  For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f:
\mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric
configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A
complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it
contains a Zariski-dense set of PCF maps. Here we prove that there are only
finitely many positive-dimensional irreducible PCF-special subvarieties in
$M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and
$B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of
degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at
most $N$ irreducible components, each with degree $\leq B$. We also prove
generalizations of these results for points with small critical height in
$M_d(\bar{\mathbb{Q}})$.
