Mycielski among trees

The two‐dimensional version of the classical Mycielski theorem says that for every comeager or conull set X⊆[0,1]2 there exists a perfect set P⊆[0,1] such that P×P⊆X∪Δ. We consider a strengthening of this theorem by replacing a perfect square with a rectangle A×B, where A and B are bodies of some types of trees with A⊆B. In particular, we show that for every comeager Gδ set G⊆ωω×ωω there exist a Miller tree TM and a uniformly perfect tree TP⊆TM such that [TP]×[TM]⊆G∪Δ and that TP cannot be a Miller tree. In the case of measure we show that for every subset F of 2ω×2ω of full measure there exists a uniformly perfect tree TP⊆2