Containments in families with forbidden subposets

We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is $O(n\binom{n}{\lfloor n/2\rfloor})$ and we find the exact value for the butterfly poset and the $N$ poset. Also, we determine the asymptotics of the maximum number of pairs in containment for some posets of which the Hasse diagram is a path.