The area immediately under a tree is by definition in shade from the branches and dry because it is protected from rainfall. In addition there will only be a thin layer of soil above the roots of the tree so selecting plants that will grow in these conditions needs some care. This article by Barb Mrgich which I found on the Master Gardeners website suggests several plants that will fit the bill.
Most of us who live in suburban areas, and try to plant a garden under a tree, meet limited success. The reason our garden doesn’t grow well can usually be attributed to dry shade. What exactly is dry shade?
Have you ever been outside when a sudden shower has you running for cover? If you run under a tree, you notice you don’t get nearly as wet. You are benefiting from the umbrella-like canape of the tree. Good for you, but bad for the plants trying to grow there. Add to this the thirsty roots of that same tree drinking up available water before the smaller plants have a chance, and you begin to understand what may be the problem with your struggling garden.
Adding organic matter to the soil is one of the best remedies for almost any gardening problem including this one. But, be careful. Piling soil on the ground under a tree can smother its roots and kill the tree. Likewise, excessive digging under a tree can severely damage its root system.
You can, however, safely add a two to three inch layer of compost or mushroom soil to the soil beneath the tree. Then select small plants, so it is not necessary to dig deep holes. That way you will disturb the tree roots as little as possible. Keep the garden well watered, and the next year add three more inches of soil amendment.
See more including the list of plants at The Master Gardeners