Welcome to JA.
I am sorry to hear about your situation.
Please note that this is not a chat service therefore you may experience delays but i will respond within 24 hours.
The law states that if your tree causes damage and or your branches overhangs onto his property then he arguably has a cause of action against you.
That said if there is no evidence of this, then i suggest you warn him to stop such actions and if not you will apply to the local county court for a civil injunction. This is a court remedy that orders the man to stop such conduct and failing to do so can amount to a criminal offence.
However, before you do this, send the neighbour a cease and desist letter to warn him of your intentions unless he stops.
Please give the neighbour 14 days to stop the conduct.
You can obtain a “cease and desist” letter template from https://www.lawdepot.co.uk/. That said you can a draft letter by your self too but ensure you make it known it is a cease and desist letter and he stops the conduct(ensure you spell out the conduct too) in question. Further it will be wise to remind him that his conducts falls within the definition given within Section 1 of the Protection from Harassment Act 1997 which is as unwanted conduct designed to cause the victim alarm or distress”. Hopefully the letter may well be enough to stop the conduct as an injunction is a serious legal remedy.
Assuming the conduct continues after you send the letter then it would be up to you whether you want to apply for the injunction. My advice be to instruct a local solicitor to assist you with completing the relevant forms. Alternatively you can return to JA and open a new question and i or one of the other experts can assist you with regards ***** ***** the relevant forms.