  In this paper we introduce a particular semigroup transform $\mathcal{A}$
that fixes the invariants involved in Wilf's conjecture, except the embedding
dimension. It also allows one to arrange the set of not ordinary and not
irreducible numerical semigroups in a family of rooted trees. We study also
another transform, having similar features, that has been introduced by
Bras-Amor\'os, and we make a comparison of them. In particular we study the
behaviour of the embedding dimension under the action of such transforms,
providing some consequences concerning Wilf's conjecture.
