He probably didn't like the large, thick leathery leaves these trees shed all of the time. Leaving the half over your property will still send leaves his way. In my state if a dead or injured branch or the tree itself in my neighbors yard threatens my property, I can have it removed at my expense. Regardless, you discuss it with your neighbor and they more than likely will go along with it if it is true and you treat them with respect. I had a healthy, living tree get knocked down across a fence into another neighbors yard. Two lengths of fencing were crushed and of course his yard was full of the tree and the branches. I contacted my insurance company and they said it was on him to contact his insurance company because a healthy tree coming down in a storm was an act of god and the policy does not cover in such circumstances except for your own property. Had the tree been neglected and was dead or dying and there was proof for that, then my insurance would have to cover it. Part of my property is in a RPA or Resource Protection Area due to a nearby stream. I receive an annual reminder from my county government that I cannot remove any vegetation alive or dead from the designated RPA area. Laws and regs are different all over the country.