On generating functions of finitely presented operads
~~ It is well known that the generating function of a generic finitely presented algebra is rational.
The purpose of this talk is to present an answer on a similar question in the case of algebraic operads.
Namely, I will show that the generating series of a generic nonsymmetric operad is an algebraic function
and the generating series of a generic symmetric operad is differentially algebraic.
Despite the motivation coming from the operad theory, a substantial part of the talk will only deal with the avoidance problems for (labeled) rooted trees hence will be accessible to nonspecialists.
based on joint work with D.Piontkovsky (arXiv:1202.5170).