If you sue the tree surgeons for negligence, their insurers will sue your neighbours for *can't be bothered to think through the exact cause of action but there will be one - negligence or breach of implied term in their contract, probably*, which will have one and the same effect.
Seriously, leave it a few days to calm down, and then do try to talk it through before sending letters, let alone solicitors letters. Do check the boundary position (often stated in the deeds) first so you know you are on solid ground as it were.
Really, really, really not the sort of thing to litigate over, good though it always sounds. I say that as a litigator (albeit not of this sort of stuff). You can request/insist/demand all sorts of things to get them to make amends at a neighbourly level without invoking laws and lawyers.