be obtained as a sum of nodes in a finite subtree $T'$ of $T$ rooted in $\textbf{n}_i^j$.

Notice that from these properties it follows that we must have multiplicity sequences along each branch. Therefore a multiplicity tree $T$ of an Arf semigroup of $\mathbb{N}^r$ can be represented by an ordered collection of $r$ multiplicity sequences $E$ and by an upper triangular matrix $ r \times r$ $$M(T)_{E}=\left(  \begin{matrix} 0 & p_{1,2} & p_{1,3} & \ldots & p_{1,r} \\ 
0 & 0 & p_{2,3} & \ldots & p_{2,r} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots& p_{r-1,r} \\ 0 & 0 & 0 & \ldots & 0 \\  \end{matrix}\right)$$where $p_{i,j}$ is the highest level such that the $i$-th and the $j$-th branches are glued in $T$. Suppose that $E=\left\{M_1,\ldots,M_r \right\}$, then in [@Zito Proposition 1.2], it is shown that the values that can be assigned to the $p_{i,j}$, in order to have a tree compatible with the previous properties, depend only on $M_i$ and $M_{j}$. In particular, if $M_i=M_{j},$ $p_{i,j}$ can be any positive integer, while, if $M_i \neq M_{j}$, there exists a level $k_E(i,j) \in \mathbb{N}^*$ such that $p_{i,j}$ is 