  The maximum size, $La(n,P)$, of a family of subsets of $[n]=\{1,2,...,n\}$
without containing a copy of $P$ as a subposet, has been intensively studied.
  Let $P$ be a graded poset. We say that a family $\mathcal{F}$ of subsets of
$[n]=\{1,2,...,n\}$ contains a \emph{rank-preserving} copy of $P$ if it
contains a copy of $P$ such that elements of $P$ having the same rank are
mapped to sets of same size in $\mathcal{F}$. The largest size of a family of
subsets of $[n]=\{1,2,...,n\}$ without containing a rank-preserving copy of $P$
as a subposet is denoted by $La_{rp}(n,P)$. Clearly, $La(n,P) \le La_{rp}(n,P)$
holds.
  In this paper we prove asymptotically optimal upper bounds on $La_{rp}(n,P)$
for tree posets of height $2$ and monotone tree posets of height $3$,
strengthening a result of Bukh in these cases. We also obtain the exact value
of $La_{rp}(n,\{Y_{h,s},Y_{h,s}'\})$ and $La(n,\{Y_{h,s},Y_{h,s}'\})$, where
$Y_{h,s}$ denotes the poset on $h+s$ elements $x_1,\dots,x_h,y_1,\dots,y_s$
with $x_1<\dots<x_h<y_1,\dots,y_s$ and $Y'_{h,s}$ denotes the dual poset of
$Y_{h,s}$.
