The expected number of symmetries in locally restricted trees II

Isomorphism and Symmetries in Random Phylogenetic Trees

“…In the course of our investigations on analytic properties of the bivariate generating function F (z, u), we came up with a few additional estimates, which improve on those of [15].…”

“…Our starting point parallels Lemmas 1-2 of [15], but we need a specific argument for the upper bound. (ii) Upper bound.…”

“…[5], [11] and [16].) In the case of U n , the expectation of the cardinality of this group has been determined by McKeon [15] to grow roughly as 1.336 09 n . In the case of phylogenetic trees (B n ), we find an expected growth of the rough form 1.241 62 n , where the exponential rate 1.241 62 .…”

“…The coefficients constitute sequence A003609 in [19]. Interested readers may consult McKeon's studies [14], [15] for details. The first few elements of the sequence are 1, 2, 2, 10, 14, 42, 90, 354.…”

“…For u in a small complex neighborhood of 1 2 , the generating function z → F (z, u) has a dominant singularity ρ(u) that is an analytic solution of (15) and is such that ρ( 2 ). As a consequence, estimates that parallel those of (16) and (17) are seen to hold, but with u ∈ now near 1 2 .…”