Ummm, original question. It's hard to disinfect trees and dirt/rocks/whatever is going to be on the ground. You need to have good drainage so you can hose down the rocks/whatever under the tree to wash off the bird poop. Most zoos replace the top however many inches of dirt in their enclosures at least once a year because of that disinfection nightmare. At a minimum, keep the birds in a conventional cage and ensure that they're disease and parasite free before turning them loose in the tree area.
Just some random thoughts:
If doing a cage outside:
I would lay down a layer of cement in the bottom and have it grooved like they do for dairy barns (the cement is poured, then while it's still wet, you "sweep" it in long strokes with a push broom all in one direction...or you can have it mechanically done after it's dried). This makes it easier to clean than a stone layer, and non-slip. Or you can buy the ceramic floor tiles that are textured like cobblestone (saw some at Home Depot and thought they would be excellent for a fish room...hehehe) and lay them instead. They both can be swept out or washed down with the water hose if needed.
You can always do what we did at the kennel I worked at if you need to clean more vigorously. Go to your local garden center and get a liquid fertilizer container that attaches to your water hose (like the ones you see in the commercials for Miracle Grow). You want a new clean one that doesn't have pre-measured anything it it. Fill it with a 50% bleach solution and attach it to your hose. Spray down the area well, let sit for 5 minutes and then rinse throughly. This gives the bleach plenty of time to kill the bacteria and de-nature the viruses (I knew I'd use this biology degree for something

Go microbiology! ). After waiting that time the feces and dried foods should be soft so take a stiff bristled broom and scrub it down. Rinse again really well.
This would only be good of a really deep cleaning (i.e. a sick bird, or something like "spring cleaning"). Bleach odors are hard on bird's lungs, so you
would have to remove the birds before doing this. After rinsing, the chlorine will dissipate, just like it does out of aged tap water. I also would not use it on anything porous, like wood.
As for inside:
Think about using reptile carpet (Astroturf with very short fibers) as long as you are going to have finches and the like (i.e. birds that wouldn't pull anything apart). This would be a bad idea with parrots since they will destroy anything not made of concrete and steel. You can just roll it up and carry it outside for a cleaning with the water hose. It comes in a bunch of colours, so you don't just have to have green. And it's cheap. If the section you have gets nasty, just but a roll and cut a new piece. If the cage is near a window you could try to grow grass in a dirt tray, but that would have to be replaced often.
You could try making your own small, hollow cement tree like some of the zoos/private collections have. I have an idea how it's done and can probably find a link for you if you want. This could be cleaned with a stiff brush and some hot water/disinfectant like Novalsan (not sure who makes this now, but it used to be made by Novartis and is used in zoos and vet's offices for sanitation of enclosures and equipment).
One last thing:
This is one of the coolest sites for custom enclosures that I have ever seen. They are pricey, but from the people I have talked to they are the best for cleaning and for looks to go in a house. They become a piece of furniture, not some ugly stainless steel thing in the corner...
http://www.cagesbydesign.com/
equus_peduus- Thanks for the kestrals/falcon clear-up. I couldn't remember from my ornithology lessons which section they belonged.

You seem very knowledgeable about animals in general, do you work with them?