  Two-dimensional version of the classical Mycielski theorem says that for
every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set
$P\subseteq [0,1]$ such that $P\times P\subseteq X\cup \Delta$. We consider
generalizations of this theorem by replacing a perfect square with a rectangle
$A\times B$, where $A$ and $B$ are bodies of other types of trees with
$A\subseteq B$. In particular, we show that for every comeager $G_\delta$ set
$G\subseteq \omega^\omega\times \omega^\omega$ there exist a Miller tree $M$
and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq
G\cup\Delta$ and that $P$ cannot be a Miller tree. In the case of measure we
show that for every subset $F$ of $2^{\omega}\times 2^\omega$ of full measure
there exists a uniformly perfect tree $P\subseteq 2^{<\omega}$ such that
$[P]\times[P]\subseteq F\cup\Delta$ and no side of such a rectangle can be a
body of a Silver tree or a Miller tree. We also show some properties of forcing
extensions of the real line from which we derive nonstandard proofs of
Mycielski-like theorems via Shoenfield Absoluteness Theorem.
