  A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal F$ is a copy of the
poset $P$ if there exists a bijection $i:P\rightarrow
\{F_1,F_2,\dots,F_{|P|}\}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. A
family $\mathcal F$ is $P$-free, if it does not contain a copy of $P$. In this
paper we establish basic results on the maximum possible number of $k$-chains
in a $P$-free family $\mathcal F\subseteq 2^{[n]}$. We prove that if the height
of $P$, $h(P) > k$, then this number is of the order
$\Theta(\prod_{i=1}^{k+1}\binom{l_{i-1}}{l_i})$, where $l_0=n$ and $l_1\ge
l_2\ge \dots \ge l_{k+1}$ are such that $n-l_1,l_1-l_2,\dots,
l_k-l_{k+1},l_{k+1}$ differ by at most one. On the other hand if $h(P)\le k$,
then we show that this number is of smaller order of magnitude.
  Let $\vee_r$ denote the poset on $r+1$ elements $a, b_1, b_2, \ldots, b_r$,
where $a < b_i$ for all $1 \le i \le r$ and let $\wedge_r$ denote its dual. For
any values of $k$ and $l$, we construct a $\{\wedge_k,\vee_l\}$-free family and
we conjecture that it contains asymptotically the maximum number of pairs in
containment. We prove that this conjecture holds under the additional
assumption that a chain of length 4 is forbidden. Moreover, we prove the
conjecture for some small values of $k$ and $l$. We also derive the asymptotics
of the maximum number of copies of certain tree posets $T$ of height 2 in
$\{\wedge_k,\vee_l\}$-free families $\mathcal F \subseteq 2^{[n]}$.
