  Fix $d \ge 2$ and a field $k$ such that $\mathrm{char}~k \nmid d$. Assume
that $k$ contains the $d$th roots of $1$. Then the irreducible components of
the curves over $k$ parameterizing preperiodic points of polynomials of the
form $z^d+c$ are geometrically irreducible and have gonality tending to
$\infty$. This implies the function field analogue of the strong uniform
boundedness conjecture for preperiodic points of $z^d+c$. It also has
consequences over number fields: it implies strong uniform boundedness for
preperiodic points of bounded eventual period, which in turn reduces the full
conjecture for preperiodic points to the conjecture for periodic points.
