We investigate two closely related tree-like forcing notions for adding reals, and heir corresponding ideals. Both forcing notions are non-ccc, yet have many properties similar to Cohen forcing (for example, they force the same values of the cardinal characteristics in Cichon's diagram). Also, the notions of regularity derived from these trees are closely related to the Baire Property, yet are subtly different, which makes this a very interesting case study. We also consider some cardinal characteristics related to these trees. Finally, we look at the connection to Fremlin's problem about “adding half a Cohen real without adding a Cohen real”, which was recently solved by Zapletal.
This is joint work with Giorgio Laguzzi.