  We prove an analog of the famous equidistribution theorem of Brolin for
rational mappings in one variable defined over the p-adic field C_p. We
construct a mixing invariant probability measure which describes the asymptotic
distribution of iterated preimages of a given point. This measure is supported
on the Berkovich space associated to the projective line over C_p. We show that
its support is precisely the Julia set as defined by Rivera-Letelier. Our
results are based on the construction of a Laplace operator on real trees with
arbitrary number of branching as done by Favre-Jonsson.
