A Bijection for Enriched Trees

Let R and F be two species, and let AR be the species of R-enriched trees. G. Labelle obtained a combinatorial proof of the fact that F(AR)[n] has the same cardinality as F′Rn[n -1] for any n ⩾ 1, where F′ is the derivative of F and [n] denotes the set {1, 2,…, n}. This leads to a combinatorial proof of the classic Lagrange inversion formula. In this paper, we present a clear-cut bijection between F (AR)[n] and F ′Rn[n - 1] by using our algorithm for decomposing an enriched tree into a forest of enriched small trees. Our bijection is also valid for Joyalʹs linear species version of the Lagrange inversion formula for ordinary formal power series.