  We show that the set of complex points in the moduli space of polynomials of
degree d corresponding to post-critically finite polynomials is a set of
algebraic points of bounded height. It follows that for any B, the set of
conjugacy classes of post-critically finite polynomials of degree d with
coefficients of algebraic degree at most B is a finite and effectively
computable set. In the case d=3 and B=1 we perform this computation. The proof
of the main result comes down to finding a relation between the "naive" height
on the moduli space, and Silverman's critical height.
