Answer: A number of small trees and large shrubs adapt well to container growing, as long as the containers are sufficiently large to allow reasonable root growth, and they receive adequate water. If the area you describe as getting no direct sun is very dark, I'm concerned that you'll have trouble getting anything to thrive. If, on the other hand, it receives dappled light or is reasonably bright due to reflected light, you might try holly. In the area that receives some direct sun, you have a few more choices. You might try flowering cherry, crabapple (these will both drop fruit, however), dwarf weeping birch, thornless honey locust, Callery pear, junipers, or yews.
I hope this helps.
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