  Let $\mathcal{S}$ be a fixed family of graphs on vertex set $V$ and
$\mathcal{G}$ be a collection of elements in $\mathcal{S}$. We investigated the
transversal problem of finding the maximum value of $|\mathcal{G}|$ when
$\mathcal{G}$ contains no rainbow elements in $\mathcal{S}$. Specifically, we
determine the exact values when $\mathcal{S}$ is a family of stars or a family
of trees of the same order $n$ with $n$ dividing $|V|$. Further, all the
extremal cases for $\mathcal{G}$ are characterized.
