  We study functional graphs generated by several quadratic polynomials, acting
simultaneously on a finite field of odd characteristic. We obtain several
results about the number of leaves in such graphs. In particular, in the case
of graphs generated by three polynomials, we relate the distribution of leaves
to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also
present extensive numerical results which we hope may shed some light on the
distribution of leaves for larger families of polynomials.
