Are you needing something to grow under the tree or at tree drip line?? Asparagus has a huge root system and needs to be cut back after frost. It needs sun so under the tree wouldn't work. If you don't want something tall then again it wouldn't work. but for a perennial, good root system and wonderful tasting vegetable to boot about all I can think of is this or lilies. There's one other idea it is a grass but I hate it myself. It grows in clumps and about kills the lawnmower when I mow over it. The grass is called orchard grass.. It too has a massive root system.

If you are wanting something to hold down the soil, is shorter in size, and a perennial, I would go with clover or vetch. It's invasive though, so I would not plant it where it needs to keep in a small space. You can also try to contain it by building a little edging on the perimeter of your grass lawn with stones or metal edging material. Wildflowers are also a good option. They will usually thrive without care and come back every year.

I can attest Vetch is VERY INVASIVE! I can't get rid of it and I am trying.