local semigroups (cf. [@BDF Theorem 2.5]).

An Arf semigroup of $\mathbb{N}^r$, is a good semigroup such that $ S(\alpha)-\alpha $ is a semigroup, for each $\alpha \in S$ , where $ S(\alpha)=\left\{\beta \in S; \beta \geq \alpha \right\}$. The multiplicity tree $T$ of a local Arf semigroup $S\subseteq \mathbb{N}^r$ is a tree where the nodes are vector $ \textbf{n}_i^j \in \mathbb{N}^r$, where with $\textbf{n}_i^j$ we mean that this node is in the $i$-th branch on the $j$-th level (the root of the tree is $\textbf{n}_{1}^1=\textbf{n}_i^1 $ for all $i$ because we are in the local case and at level one all the branches must be glued) and we have

$$S=\left\{\textbf{0}\right\} \bigcup_{T'} \left\{ \sum_{\textbf{n}_i^j \in T' } {\textbf{n}_i^j}\right\},$$

where $T'$ ranges over all finite subtree of $T$ rooted in $\textbf{n}_1^1$.

Furthermore a tree $T$ is a multiplicity tree of an Arf semigroup if and only if its nodes satisfy the following properties (cf. [@BDF Theorem 5.11] ).

-   There exists $L \in \mathbb{N}$ such that for $m \geq L$, $\textbf{n}_i^m=(0,\ldots,0,1,0\ldots,0)$ (the nonzero coordinate is in the $i$-th position) for any $i=1,\ldots,n$;

-   $\textbf{n}_i^j[h]=0$ if and only if $\textbf{n}_i^j$ is not in the $h$-th branch of the tree;

-   Each $\textbf{n}_i^j$ can 