Acyclic graphs with at least $2\mathrm{\ell }+1$ vertices are $\mathrm{\ell }$-recognizable

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