On the number of quadratic polynomials with a given portrait

Abstract: Let $F$ be a number field. Given a quadratic polynomial $f_c(z) = z^2 + c \in F[z]$, we can construct a directed graph $Preper(f_c, F)$ (also called a portrait), whose vertices are $F$-rational preperiodic points for $f_c$, with an edge $\alpha \to \beta$ if and only if $f_c(\alpha) = \beta$. Poonen and Faber classified the portraits that occur for infinitely many $c$'s. Given a portrait $P$, we prove an asymptotic formula for counting the number of $c \in F$'s by height, such that $Preper(f_c, F) \cong P$. We also prove an asymptotic formula for the analogous counting problem, where $Preper(f_c, K) \cong P$ for some quadratic extension $K/F$. These results are conditioned on Morton-Silverman conjecture.