Categorification of Hopf algebras of rooted trees

We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec ?) whose semiring of functions is (a P-version of) the Connes-Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to ? and collapsing H ₀). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.