  We establish the equidistribution with respect to the bifurcation measure of
post-critically finite maps in any one-dimensional algebraic family of
unicritical polynomials. Using this equidistribution result, together with a
combinatorial analysis of certain algebraic correspondences on the complement
of the Mandelbrot set $M_2$ (or generalized Mandelbrot set $M_d$ for degree
$d>2$), we classify all complex plane curves $C$ with Zariski-dense subsets of
points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously
post-critically finite for a fixed degree $d\geq 2$. Our result is analogous to
the famous result of Andre regarding plane curves which contain infinitely many
points with both coordinates CM parameters in the moduli space of elliptic
curves, and is the first complete case of the dynamical Andre-Oort phenomenon
studied by Baker and DeMarco.
