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You have a common law right to cut back tree branches that overhang onto your property. It is however always best to discuss with your neighbour about any trees / hedges you wish to cut back before doing so.
You are only entitled to cut the branches by being positioned on your land and only entitled to cut any branches from the point on the tree that overhangs the boundary line.
Hopefully your tree surgeon will be fully adverse in what he can and can not do with the branches, but hope this assists you along the way.
Thanks for your reply.
He should ask either for the return of all of them or none of them!