higher dimensions. In order to do that we recall the definition of good semigroups and the properties of the multiplcity tree of an Arf local good semigroup and we accordingly generalize the concept of genus to good semigroups of $\mathbb{N}^r$. We define the sets $\textrm{Gen}(r,n)$ of the untwisted multiplicity trees of all the Arf good semigroups of $\mathbb{N}^r$ with genus $n$. In Theorem \[genere\] we give a way to compute the genus of an untwisted Arf good semigroup of $\mathbb{N}^r$ from its representation $T_E$. Using this theorem it is possible to develope, using induction on $r$, a procedure that calculate the set $\textrm{Gen}(r,n)$, and from Proposition \[raff\] we deduce a possible way to make this computation faster. We give a strategy for computing the set $\overline{\textrm{Gen}(r,n)}$ of all the possible multiplicity trees (twisted and untwisted) associated to an Arf semigroup with genus $n$ in $\mathbb{N}^r$. At the end of the section we give an example of the application of the developed procedure and we present some tables containing cardinalites of the sets $\textrm{Gen}(r,n)$ for some values of $r$ and $n$.

The procedures presented here have been implemented in GAP ([@gap]).

Arf good semigroups of $\mathbb{N}^2$ with given conductor {#section2}
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