If you're anticipating issues with the neighbor, give your local municipality a shout. There are very likely rules that will dictate who is allowed to do what, when, and if there's any notification necessary.

For instance, your borough/whatever may have ordinances in place that allow any homeowner the right to remove tree limbs hanging over their property, if they're in danger of causing damage. If that's the case, you should be completely within your rights to have the tree pruned without needing to involve the neighbor - but check with your local borough manager or whomever, first.

At the very least, they'll already be aware that you're trying to be a good community member IF/WHEN the evil neighbor tries to raise a stink about it.