  We establish a quantitative criterion for an operator defined on a
Galton-Watson random tree for having an absolutely continuous spectrum. For the
adjacency operator, this criterion requires that the offspring distribution has
a relative variance below a threshold. As a byproduct, we prove that the
adjacency operator of a supercritical Poisson Galton-Watson tree has a
non-trivial absolutely continuous part if the average degree is large enough.
We also prove that its Karp and Sipser core has purely absolutely spectrum on
an interval if the average degree is large enough. We finally illustrate our
criterion on the Anderson model on a d-regular infinite tree with d $\ge$ 3 and
give a quantitative version of Klein's Theorem on the existence of absolutely
continuous spectrum at disorder smaller that C $\sqrt$ d for some absolute
constant C.
