of $T$ rooted in it. In [@Zito], taking in account this $1$-$1$ correspondence, it is showed a way to describe a multiplicity tree of an Arf semigroup of $\mathbb{N}^r$ by mean of an ordered collection $E$ of multiplicity sequence and a matrix $M(T)_E=(p_{i,j})$. In the same paper it is shown that we can consider a particular class of trees, the untwisted ones, that are easier to study because they can be expressed by a vector of $\mathbb{N}^{r-1}$ instead of a matrix. This is not restrictive because every multiplicity tree can be transformed by an opportune permutation into an untwisted one (this implies that the corresponding semigroups are isomorphic).

The conductor $\textbf{c}$ of a good semigroup is the minimal vector such that $ \textbf{c}+ \mathbb{N}^r \subseteq S$ (the existence of such a vector is guaranteed by the properties of good semigroups). In [@Zito2], it is shown how to deduce the value of the conductor of an Arf semigroup from the representation of its multiplicity tree and the author presented procedures for the computation of all Arf semigroups of $\mathbb{N}^r$ with a prescribed conductor. In order to generalize the procedure introduced in the numerical case, we need to extend the concept of 