  An unusual and surprising expansion of the form \[ p_n = \rho^{-n-1}(6n
+\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller
order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that
two randomly chosen binary search trees are identical (in shape and in labels
of all corresponding nodes). A quantity arising in the analysis of phylogenetic
trees is also proved to have a similar asymptotic expansion. Our method of
proof is new in the literature of discrete probability and analysis of
algorithms, and based on the psi-series expansions for nonlinear differential
equations. Such an approach is very general and applicable to many other
problems involving nonlinear differential equations; many examples are
discussed and several attractive phenomena are discovered.
