of one dimensional analytically unramified ring (for example the local ring of an algebraic curve), although it is possible to show that the resulting class of good semigroups is bigger than the class of value semigroups. Therefore the good semigroups can be seen as a natural generalization of the numerical semigroups. A good semigroup $S \subseteq \mathbb{N}^n$ is said to be local if the zero vector is the only element of $S$ with zero coordinates.

Then we can give the definition of Arf semigroup in the context of good semigroups of $\mathbb{N}^r$, by adapting in a natural way the numerical version, considering the usual partial ordering of $\mathbb{N}^r$. Furthermore, in order to extend the concept of multiplicity sequence, in [@BDF] it is also shown that to each local Arf good semigroup, it can be associated a multiplicity tree that characterizes the semigroup completely. A tree $T$ of vectors of $\mathbb{N}^n$ has to satisfy some properties to be a multiplicity tree of a local Arf good semigroup. For instance, it must have multiplicity sequences along its branches (because its projections are Arf numerical semigroups) and each node must be able to be expressed as a sum of nodes in a subtree 