  We study invariant percolation processes on the d-regular tree that are
obtained as a factor of an iid process. We show that the density of any factor
of iid site percolation process with finite clusters is asymptotically at most
(log d)/d as d tends to infinity. This bound is asymptotically optimal as it
can be realized by independent sets. One implication of the result is a
(1/2)-factor approximation gap, asymptotically in d, for estimating the density
of maximal induced forests in locally tree-like d-regular graphs via factor of
iid processes.
