The value semigroup of a plane curve singularity with several branches

We present a constructive procedure, based on the notion of Apéry set, to obtain the value semigroup of a plane curve singularity from the value semigroup of its blow-up and vice-versa. In particular, we give a blow-down process that allows us to reconstruct a plane algebroid curve form its blow-up, even if it is not local. Then, we characterize numerically all the possible multiplicity trees of plane curve singularities, obtaining in this way a constructive description of all their value semigroups.