  We extend classical results on simple varieties of trees (asymptotic
enumeration, average behavior of tree parameters) to trees counted by their
number of leaves. Motivated by genome comparison of related species, we then
apply these results to strong interval trees with a restriction on the arity of
prime nodes. Doing so, we describe a filtration of the set of permutations
based on their strong interval trees. This filtration is also studied from a
purely analytical point of view, thus illustrating the convergence of analytic
series towards a non-analytic limit at the level of the asymptotic behavior of
their coefficients.
