\textrm{Cond}^*(4) $ and $E_8=\left\{ M_1,M_2 \right\}$ we have $$l(M_1)+c[1]-k_1=1+4-3=2=1+5-4=l(M_2)+c[2]-k_2\leq \textrm{Comp}(M_1,M_2)=4.$$ Thus we have the tree $T_9=T_{E_8}=(2).$ Hence $S^3([4,5])=\left\{ T_8,T_9\right\}$. 0.3in

    \]at(current bounding box.south)[$T_6$]{};

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    \]at(current bounding box.south)[$T_7$]{};

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    \]at(current bounding box.south)[$T_8$]{};

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    \]at(current bounding box.south)[$T_9$]{};

Summarizing we have $ \textrm{Cond}([4,5])=\left\{ T_1,T_2,T_3,T_4,T_5, T_6,T_7,T_8,T_9\right\}$.

Using the previous results it is easy to implement an algorithm that computes the number of Arf semigroups of $ \mathbb{N}^2$ with a given conductor. Each entry of the following table is such a number, where the conductors range from $(1,1)$ to $(20,20)$.

Arf semigroups of $\mathbb{N}^r$ with a given conductor {#section3}
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In this section we study the general case. We want to develope a recursive procedure to calculate $\textrm{Cond}(\textbf{c})$ for $\textbf{c} \in \mathbb{N}^r$, using the fact that we already know how to solve the base cases $r=1$ and $r=2$. In order to do that is very useful the following Lemma.

\[Lemma3\] Consider $\textbf{c}=(c[1],\ldots,c[r]) \in \mathbb{N}^r$ and suppose that $T=T_{E}=(p_1,\ldots,p_{r-1}) \in \textrm{Cond}(\textbf{c})$ where $E=\left\{ M_1,\ldots,M_r \right\}$. If $t \in \left\{ 2,\ldots,r-1\right\}$, then we have that at least one of the following conditions must hold:

-   $ T_1= T_{E_1}=(p_1,\ldots,p_{t-1}) \in \textrm{Cond}((c[1],\ldots,c[t]))$ with $E_1=\left\{M_1,\ldots,M_t\right\};$

-   $T_2= T_{E_2}=(p_t,\ldots,p_{r-1}) \in \textrm{Cond}((c[t],\ldots,c[r]))$ with $E_2=\left\{M_t,\ldots,M_r\right\}.$

**Proof**. We assume by contradiction that

-   $ 