You haven't actually stated what you want any new plantings/tree need to accomplish for you; give privacy from views by neighboring apartment building?soften the look of the wall? Be a garden setting that tolerates the conditions? Serve as a garden area for particular functions?

Beyond that, when seeking planting advice on this site, it also helps to know what level of maintenance you care to provide, how big you want things to get, whether you prefer evergreen versus deciduous, etc.

Oops, guess I thought the pic said it all. This is actually just an added pic to the main post. The original, with first pic, went into more detail as to deciduous, zone, light. ( please see the main post, " I'm so ashamed") The neighbors house with balcony is uphill from my house. I'm unsure what tree will thrive in very little full sun. As you can see, it is not dark shade. Can probably imagine the needed height: approx. 20 feet or so.

There must be understory trees in your area that would mind the lighter shade. Ie trees that often will be smaller and grow under larger trees. Not sure what grows in your area but that is what I'd look for. Maybe go to a local nursery and ask what they would recommend.