Transversals in a collections of trees

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.