  Given a rational function of degree at least two defined over a number field
k, we study the cardinality of the set of rational iterated preimages. We prove
bounds for the cardinality of this set as the rational function varies in
certain families. Our proofs are based on unit equations and a method of Runge
for effectively determining integral points on certain affine curves. We also
formulate and state a uniform boundedness conjecture for iterated preimages of
rational functions and relate this conjecture to other well-known conjectures
in arithmetic dynamics.
