This is one of those problems that would be better solved before you take the tree down. Perhaps having a third-party (like a lawyer) contact the neighbor and trying to negotiate the tree removal would work. If not, you are left with the options of leaving the tree up, or cutting it down and dealing with the consequences later. Those consequences could be nothing - or the neighbor may sue you for trespassing. In such a case, the damages would be something the court would determine.