Anti-Ramsey forbidden poset problems

A family $\mathcal{G}$ of sets is a weak copy of a poset $P$ if there is a bijection $f:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ implies $f(p)\subseteq f(q)$. If $f$ satisfies $p\leqslant q$ if and only if $f(p)\subseteq f(q)$, the $\mathcal{G}$ is a strong copy of $P$. We study the anti-Ramsey numbers $\mathrm{ar}(n,P), \mathrm{ar^*}(n,P)$, the maximum number of colors used in a coloring of $2^{[n]}$ that does not admit a rainbow weak or strong copy of $P$, respectively. We establish connections to the well-studied extremal numbers $\mathrm{La}(n,P)$ and $\mathrm{La^*}(n,P)$ and determine asymptotically $\mathrm{ar^*}(n,T)$ for all tree posets $T$ and $\mathrm{ar^*}(n,O_{2k})$ for all crown posets $O_{2k}$.