I understand that can get old and annoying. You did the right thing by calling him and asking him to repair it, and he said that he will. Chances are good though that he won't take the trees or limbs down. You may find yourself calling him a few more times in years to come. I'm not saying that you should have to, only pointing out that you may find yourself doing so.
Back in the late '70's, North Korea had a tree on the border with limbs overhanging and causing problems for South Korea by blocking something, although I don't recall what it was blocking. Truth be told, North Korea had troops near by and was daring South Korea to trim the limbs in a provocation. Enough was enough, so we sent troops from camp to the site and cut down North Korea's tree right in front of them and dared them. They packed up and left. Different circumstances require different solutions. You did the right thing, and so did we. How things get handled depends upon the other guy, for the record.