I have the same problem under my Birch Trees. I am thinking about some large colorful flower pots for container gardens. Hostas would be great. Also you could add a few of those mini Coral Bells, Corydalis, Bugleweed, and maybe some small grasses.
I have been racking my brain for a long time to come up with a solution for my problem. I read that if you do a raised bed that close to the trees you will suffocate the roots because they need air. So the large colorful containers would be the easiest thing I think. Also will brighten things up a lot and will not compete for water. Oh, don't forget to add some painted ferns.
If you are in zone 9 be sure to choose heat tolerant Hostas. Read this article, you will like it.
http://www.plantdelights.com/H...
