Bilabelled increasing trees and hook-length formulae

We introduce two different kind of increasing bilabellings of trees, for which we provide enumeration formulæ. One of the bilabelled tree families considered is enumerated by the reduced tangent numbers and is in bijection with a tree family introduced by Poupard [11]. Both increasing bilabellings naturally lead to hook-length formulas for trees and forests; in particular, one construction gives a combinatorial interpretation of a formula for labelled unordered forests obtained recently by Chen et al. [1].