A census of quadratic post-critically finite rational maps defined over   Q
  We find all quadratic post-critically finite (PCF) rational maps defined over the rationals. We describe an algorithm to search for possibly PCF maps. Using the algorithm, we eliminate all but twelve rational maps, all of which are verifiably PCF. We also give a complete description of possible rational preperiodic structures for quadratic PCF maps defined over Q. 
David Lukas, Michelle Manes, and Diane Yap
math.NT