We consider a combinatorial question about searching for an unknown ideal $μ$ within a known pointed poset $λ$. Elements of $λ$ may be queried for membership in $μ$, but at most $k$ positive queries are permitted. We provide a general search strategy for this problem, and establish new bounds (based on $k$ and the degree and height of $λ$) for the total number of queries required to identify $μ$. We show that this strategy performs asymptotically optimally on the family of complete $\ell$-ary trees as the height grows.