  We provide a complete classification of possible graphs of rational
preperiodic points of endomorphisms of the projective line of degree 2 defined
over the rationals with a rational periodic critical point of period 2, under
the assumption that these maps have no periodic points of period at least 7. We
explain how this extends results of Poonen on quadratic polynomials. We show
that there are 13 possible graphs, and that such maps have at most 9 rational
preperiodic points. We provide data related to the analogous classification of
graphs of endomorphisms of degree 2 with a rational periodic critical point of
period 3 or 4.
