Mycielski among trees

The two-dimensional version of Mycielski theorem says that for every comeager or conull subset G of the real plane there exists a perfect subset of the real line such that the square of this perfect set P can be inscribed into set G modulo diagonal. We consider a strengthening of this theorem by replacing a perfect square with a rectangle with bodies of some type trees. In particular, we show that for every dense G-delta subset of the Cartesian product of the Baire space there is a Miller tree and uniformly perfect tree for which the cross-product of its bodies is a subset of G modulo diagonal. Moreover, the uniformly perfect tree is subtree of the Miller tree and the uniformly perfect tree cannot be replaced by any Miller tree.