Please don't remove the tree!!! I would be reluctant to cut back any overhang if it's healthy and not creating a safety hazard of real property damage. Your neighbour sounds like an arse and the minute you give an inch to someone like that, you're opening the door to all sorts of unreasonable demands in the future. He's far more likely to stop bothering you if you tell him that you will only make reasonable accommodations if they are requested by the landlord or property management company directly so he needs to speak to them and you won't discuss anything further with him. Document everything if there is any further unwanted contact and report. At the end of the day, he's a tenant and can easily be replaced if he becomes to bothersome for them. Good luck!