It's a Platycladus orientalis, aka Oriental arborvitae. You don't trim it - it doesn't require it and any pruning or trimming will destroy the growth habit.

It appears to have fruit in the upper right quadrant of the photo and they appear to have wings, perhaps sort of triangular in cross section. So unless I am mistaken, they aren't mulberry. The seeds sort of resemble my Halesia, but the leaves don't look quite right for that. LisaD, can you add a photo of the seeds/pods?

Flora, I just don't see Aralia spinosa as a street tree or tree for a small property . Where I've seen it it suckers like mad. I once grew one but it didn't remain as one and had to be removed. And it's a bit treacherous working around it. Here's a blog with a follow up posting from someone else that once grew it. http://usinggeorgianativeplants.blogspot.com/2014/07/giant-flower-at-last.html Zanthoxylum info : http://www.foragingtexas.com/2008/08/prickly-ashtoothache-tree.html

2nd looks like an ornamental pear (from here). But a closer shot of leaves would help.