Try creeping sedum. They like full sun and are drought-tolerant once established. However, as the tree grows, you'll need to replace with something more shade-tolerant. I wouldn't suggest Bishop's weed, lamium, or lamiastrum--they're too invasive and aggressive. How about creeping thyme? Thyme or sedum will fill in and if they get too scraggly just give 'em a haircut, they'll respond nicely.