I think Sandy is saying that there isn't much soil depth under the trees, because the roots of the trees are close to the surface.
Like Tyler said, you can grow plants that do fine naturally under trees, and don't require much soil depth. Cane berries (raspberries, blackberries, salmonberries, thimbleberries) all grow well under my maples, likely because they don't have very
deep roots. Strawberries
should also do well. Mint will have a party and take over.
Nettles also seem to grow right under my maples. Blueberries, salal and oregon grape might also do well, though I usually see those growing in dead
wood.
And, you could also probably make a
raised bed under the tree to get enough soil to grow deeper plants. I made a raised bed next to my
apple trees, but those were younger. From what I understand, you don't want to over ALL of the trees roots with deep soil, as they might be deprived oxygen, but a raised bed over part of the roots would probably be fine. You can read more about my bed (and what more knowledgeable people said about it, here:
https://permies.com/t/50364/Raised-Garden-Bed-Hugel-Fruit)
I hope that helps
answer your question!