Certified GKH Gardening Expert
The best thing you can do for trees is add a couple inches of mulch under the beds out to the canopy line. Adding plants under a tree can disturb the feeder roots, which are usually within 12 inches of the surface.
If you still want to plant underneath, succulent groundcovers like sedum, ice plant, candytuft, or creeping phlox would do well if plenty of sun shines through the trees. You have to plant very carefully.