If you carefully brush high-strength Roundup onto the grass, it will kill the grass without harming the tree. Use a sponge paint brush and be careful not to let the Roundup drip onto the ground.

I wouldn't try to suffocate the grass, because in order to do it effectively, you'd possibly suffocate the tree as well.

donn is right, suffocating method can be harmful to the tree's health.

Use "Roundup" but you'll have to wait for about a week or so to till it in. The herbicide needs time to go through the system and to the roots to completely kill it. After you see some yellowing of the blades you can till it in. Mix at 2 oz. of roundup per gallon of water. This will do the job. Hope this answers your question.