Supersaturation of Tree Posets

A well-known result of Bukh shows that if P is a tree poset of height k, then any subset of the Boolean lattice F Bₙ of size at least (k-1+) n n/2 contains at least one copy of P. This was extended by Boehnlein and Jiang to induced copies. We strengthen both results by showing that for any integer q k, any family F of size at least (q-1+) n n/2 contains on the order of as many induced copies of P as is contained in the q middle layers of the Boolean lattice. This answers a conjecture of Gerbner, Nagy, Patk\'os, and Vizer in a strong form for tree posets.