I'm using deep litter in my coop and love it, but this area is way too large and I have concern about giving the tree all that nutrition. It's only a few years old and it averages 3 feet per year I think. It doesn't need help.
Sand might be my best option, I dunno. Do you get much dust when you clean? I am dubious about straw and hay based on past experience after a lot of rain.
Sand was my first thought, too. The birds will probably dig a couple spots down to the dirt for dust bathing, but overall it would probably be the best option.
If you do decide to go with something plant based, I'd stay away from straw, it will just make a bigger mess of things. Hay would be better, though more expensive. Or maybe woodchips. But the birds will eventually work it all back down to a dust anyways. All my birds are on dirt, but then I live at the end of a dirt road. Dirt is the basis of everything around here. We wash our hands with it. ;-)