The free abelian group $R(Q)$ on the set of indecomposable representations of a quiver $Q$ , over a field $K$ , has a ring structure where the multiplication is given by the tensor product. We show that if $Q$ is a rooted tree (an oriented tree with a unique sink), then the ring $R(Q)_{\rm red}$ is a finitely generated $\mathbb{Z}$ -module (here $R(Q)_{\rm red}$ is the ring $R(Q)$ modulo the ideal of all nilpotent elements). We describe the ring $R(Q)_{\rm red}$ explicitly by studying functors from the category ${\rm rep}(Q)$ of representations of $Q$ over $K$ to the category of finite-dimensional $K$ -vector spaces