Im over neighbours trees so thinking about the original post my response is.
Its your tree hanging over the neighbours property that's causing the problem so prune it or remove it as its your responsibility.
Cheers for the advice Bentnose; I might ignore yours and listen to the majority that responded instead.
A bit of an update. I speak to oldmate next door last week. I shake his hand, wish him a Happy New Year. I showed him how I had cut back branches with my chainsaw, asked him if he was happy with the tree.
He tells me that he never had a problem with the tree, he said his Mrs just got a bee in her bonnet and decided to do what she did. He said they both had words about what she did and she now regrets cutting the branches and throwing them over. I tell him that if there is ever an issue, let me know and I will fix it. We both shake hands, all done and dusted.
Mrs Misery however can go and get......