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Hello, I am a solicitor with 20 years experience. I will try to help you with this.
You are not legally responsible for dealing with the overhanging branches or disposing of the branches. If your neighbour chops off the overhanging branches as he is entitled to then he has to offer them to you in case you want them. If you don't want them there is no law requiring you to to dispose of them. I am assuming in this answer that there is no structural harm being caused by the tree.
They are not trespassing. A tree cannot trespass only people can.
The new tenant can chop branches back to the boundary but his obligation then is to offer you the branches, You do not have to accept them unless you want to.
They can only sue in negligence or nuisance if there is damage being caused,
Looking at your picture it looks like the sort of situation that exists in millions of neighbouring gardens up and down the country. There is no case against you.
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