Gardening under trees and on the north side of buildings isn't really a problem. There are many plants that are natively found only under an upper canopy.

The real challenge is more about water and soil. Decidous trees can pump hundreds of liters of water per day out of the ground through their transvaporative processes depleting the upper soil layer of all moisture. And many evergreens, although not water greedy, have a tendancy to drop their needles in the autumn leaving the ground more acidic than most companions like.

Here's some plants that love shade.

   Index to shade loving plants