Acyclic graphs with at least 2ℓ + 1 vertices are ℓ-recognizable

The $\left(n-\ell \right)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $\left(n-\ell \right)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs with no cycles is $\ell$-recognizable when $n\ge 2\ell +1$ (except for $\left(n,\ell \right)=\left(5,2\right)$). As a consequence, the family of $n$-vertex trees is $\ell$-recognizable when $n\ge 2\ell +1$ and $\ell \ne 2$. It is known that this fails when $n=2\ell$.