2 thoughtsOld or broken bricks can be had cheaply at masonry suppliers or a Habitat for Humanity Re-Store, if there's one nearby. If you sink them in some sand, and place them tightly together, you can form a nice curve around your tree and have minimal "grass creep."
If you want it to look more natural, clear out your bed area and edge it as deeply as you can to destroy grass/weed roots along the edge. Then bury some of that ugly (but cheap!) flexible plastic landscape edging, right down to 1" above ground level so it's not visible. Mulch the bed well, and plant groundcover plants like monkeygrass all around the inside of the edging. You can mow right up to the monkeygrass but never see the edging.
If you already have some groundcovers growing and can divide them, that part is free. If you have to buy them but can't afford them all at once, just put them in a couple at a time as you can buy them. They fill in by themselves, somewhat.
Good luck...this is what I'm about to tackle too, and the latter seems to be the lowest-maintenance, lowest-budget solution I can find!