Should this ironwood/musclewood be pruned?
I have an ironwood/musclewood (Carpinus caroliniana)that I am trying to groom to be a small shade tree. I've seen these trees all over, but I don't know a whole lot about their growth. The tree is growing like gangbusters and I'd really like to tame it down a bit, plus make it look better when eventually the trunk grows up. It has two large branches that are seriously crossing sections of the trunk, to the point where the bark is rubbing. I'm wondering if I should prune these? I want it to have a strong trunk, and I'm wondering if I don't prune, will the branches just grow together? Really the tree is getting out of hand, and it would look better pruned, so unless it just would look prettier unpruned, I think I would like to prune unless there is some compelling reason not to. I've attached photos of the whole tree and the sections to be pruned. The second shot is blurry, but I think you can get the gist of the problem.
Anyone have any experience with this tree to share? Thanks!