The law states that any part of the tree that overhangs your property, you can trim. BUT and this is the really important bit. you must return any bits you've trimmed to your neighbor.
however you must not cause malicious damage to the tree. so cutting off the roots of the tree wood kill it, not straight away it could take up to 10 years all depending on the size. and your neighbour could sue you for a replacement tree, which depending on the tree could cost 10's of thousands,
But then you can sue your neighbour for the cost of any damage this tree is doing.
But I would have a very calm and friendly chat and explain your problem, and take it from there