We study the situations when the solution to a weighted stochastic recursion
has a power law tail. To this end, we develop two complementary approaches, the
first one extends Goldie's (1991) implicit renewal theorem to cover recursions
on trees; and the second one is based on a direct sample path large deviations
analysis of weighted recursive random sums. We believe that these methods may
be of independent interest in the analysis of more general weighted branching
processes as well as in the analysis of algorithms