  The probability that two randomly selected phylogenetic trees of the same
size are isomorphic is found to be asymptotic to a decreasing exponential
modulated by a polynomial factor. The number of symmetrical nodes in a random
phylogenetic tree of large size obeys a limiting Gaussian distribution, in the
sense of both central and local limits. The probability that two random
phylogenetic trees have the same number of symmetries asymptotically obeys an
inverse square-root law. Precise estimates for these problems are obtained by
methods of analytic combinatorics, involving bivariate generating functions,
singularity analysis, and quasi-powers approximations.
