Thanks again everyone.

I received another email telling me the tree looks like a 'silver
maple' which can be nuissance (lots of feeder roots, helicopter fruit
droppings...) and maybe I should just remove it altogther.

I like the fact that it's a big tree, it is helping shield some road
noise, but we do have a pool close by so the fruit droppings would be a
problem. I live in Ottawa Canada BTW.

I think at this point I might as well just aggresively prune the small
branch without removing it completely and hope that the central branch
grows stronger.