Set families with a forbidden induced subposet

For each poset $H$ whose Hasse diagram is a tree of height $k$, we show that the largest size of a family $\cF$ of subsets of $[n]=\{1,..., n\}$ not containing $H$ as an induced subposet is asymptotic to $(k-1){n\choose \fl{n/2}}$. This extends the result of Bukh \cite{bukh}, which in turn generalizes several known results including Sperner's theorem.