i'd be leary of putting any kind of product down! you don't want to do anything to damage to the
trees!!
put down a few layers of newspaper (at least 6 sheets) and then cover with some nice mulch.
to keep the grass from coming back, i'd suggest some kind of barrier be put down to block the spread of the grass roots. there is black plastic edging (very sturdy) that you could cut to fit the area. it can be inserted into the soil all the way up to the top (which has a nice curved lip on it) so that it's not visible. it will block the roots from the grass spreading into that area again.
you could also do some kind of brick border or those paving stones. if you want that kind of look...i get the impression you don't tho
edit:
this isn't the same brand that i have...you can get the idea of what i'm talking about tho
edging stuff