Check the state statutes and your local ordinances and any HOA, but ordinarily your neighbor cannot trespass on your property or maintain a nuisance. The law controls, but my first suggestion is talk to your neighbor and ask them to remove it. You could get their written agreement to have someone remove the branch with reimbursement to you for the expense, but then you would be assuming some liability. They own the problem-they should deal with it.
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