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In short, the neighbour's tree over hangs into your garden?
You can cut down over hanging branches yourself as long as you return the foliage to the neighbours.
What you cannot do is demand that the neighbours remove it.
If you cannot do it yourself then you can use a tree surgeon to remove the branches although the neighbours probably would not be liable for the cost. That said, it would be a lot cheaper than paying a solicitor to take legal action against him.
Can I clarify anything for you?