Squarefree powers of edge ideals of forests

Let $I(G{)}^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G{)}^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G{)}^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G{)}^{[2]}$ for any forest $G$.