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We study the asymptotic number of certain monotonically labeled increasingtrees arising from a generalized evolution process. The main difference betweenthe presented model and the classical model of binary increasing trees is thatthe same label can appear in distinct branches of the tree. In the course ofthe analysis we develop a method to extract asymptotic information on thecoefficients of purely formal power series. The method is based on anapproximate Borel transform (or, more generally, Mittag-Leffler transform)which enables us to quickly guess the exponential growth rate. With this guessthe sequence is then rescaled and a singularity analysis of the generatingfunction of the scaled counting sequence yields accurate asymptotics. Theactual analysis is based on differential equations and a Tauberian argument.The counting problem for trees of size n exhibits interesting asymptoticsinvolving powers of n with irrational exponents.

On the Number of Increasing Trees with Label Repetitions