Hello, I am a solicitor with 20 years experience. I will try to answer this for you.
He had no right to cut anything off your tree except tot eh extent that it overhung his land.
He may have committed the crime of criminal damage, and you could report it to the police. They could investigate and arrest him. They may not be that interested in it though.
You are the victim of civil wrong - an act of trespass- and could possibly bring a civll action to compensate you for the harm you have suffered.
Have you experienced loss of privacy, loss of shade, or the loss of attractive tree. Has your neighbour gained anything eg increased sunlight?
What harm have you suffered loss of privacy, loss of shade, he