  We consider random subgroups of Thompson's group $F$ with respect to two
natural stratifications of the set of all $k$ generator subgroups. We find that
the isomorphism classes of subgroups which occur with positive density are not
the same for the two stratifications.
  We give the first known examples of {\em persistent} subgroups, whose
isomorphism classes occur with positive density within the set of $k$-generator
subgroups, for all sufficiently large $k$. Additionally, Thompson's group
provides the first example of a group without a generic isomorphism class of
subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite
rooted binary trees.
  We compute the asymptotic growth rate and a generating function for the
number of reduced pairs of trees, which we show is D-finite and not algebraic.
We then use the asymptotic growth to prove our density results.
