Tree Posets: Supersaturation, Enumeration, and Randomness

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $\mathcal{F}$ is a family in $\mathcal{B}_n$ with $|\mathcal{F}|\ge (q-1+\varepsilon){n\choose \lfloor n/2\rfloor}$ for some $q\ge k$, then $\mathcal{F}$ contains on the order of as many induced copies of $P$ as is contained in the $q$ middle layers of the Boolean lattice. This generalizes results of Bukh and Boehnlein and Jiang which guaranteed a single such copy in non-induced and induced settings respectively. * The number of induced $P$-free families of $\mathcal{B}_n$ is $2^{(k-1+o(1)){n\choose \lfloor n/2\rfloor}}$, strengthening recent independent work of Balogh, Garcia, Wigal who obtained the same bounds in the non-induced setting. * The largest induced $P$-free subset of a $p$-random subset of $\mathcal{B}_n$ for $p\gg n^{-1}$ has size at most $(k-1+o(1))p{n\choose \lfloor n/2\rfloor}$, generalizing previous work of Balogh, Mycroft, and Treglown and of Collares and Morris for the case when $P$ is a chain. All three results are asymptotically tight and give affirmative answers to general conjectures of Gerbner, Nagy, Patk\'os, and Vizer in the case of tree posets.