genus to good semigroups of $\mathbb{N}^r$. This can be done by using the function $d(\textunderscore\setminus \textunderscore )$ defined in [@D'a] . In particular, if $S$ is a good semigroup of $\mathbb{N}^r$ with conductor $\delta$, then $C=\delta+\mathbb{N}^r$ is an ideal of $S$ and we can define the genus of $S$ as $ g(S)=d(\mathbb{N}^r \setminus C) - d(S \setminus C). $ Then, using the aforementioned representation for multiplicity trees and the procedure developed in the numerical case, we address the problem of computing the set $\textrm{Gen}(r,n)$ of all the Arf good semigroups of $\mathbb{N}^r$ with an untwisted tree and genus $n$.

The structure of the paper is the following.

In Section \[section4\], we firstly recall some definitions and properties concerning the numerical semigroups. Then we address the problem, already studied in [@algo], of finding the set $\textrm{Gen}(n)$ of the multiplicity sequences of all the Arf numerical semigroups with genus $n$. We accordingly adapt the procedure presented in [@Zito2] for the computation of all the Arf numerical semigroups with a given conductor, presenting a non recursive algorithm for the computation of $\textrm{Gen}(n)$, that appears to be faster than the one presented in [@algo].

In Section \[section5\], we begin to deal with semigroups in 