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I am Al and am happy to assist you with your enquiry.
Unless the branches are actually causing physical damage to his property, then your neighbour has no legal grounds to demand that you trim the branches.
The mere fact that your branches are overhanging the boundary, and may be for example dropping leaves/blocking his light, does not give your neighbour any legal right to make you take any positive action- he is legally entitled entitled to lop the branches themselves (provided there is no Tree Preservation Order on it) and return the branches back over the boundary to you. If he wants the branches lopped, this is what he should do!
I hope this assists you and sets out the legal position.