The number of elements in the mutation class of a quiver of type $D_n$

We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by ${\sum }_{d\mathrm{\mid }n}\varphi (n\mathrm{/}d)\left(\genfrac{}{}{0px}{}{2d}{d}\right)\mathrm{/}(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.