If they really are only 2-3 m away I would have thought that the major problem was likely to be roots growing under the building. The building insurers are like to see it as a problem, and a letter reminding the neighbour of their personal liability for structural repairs may put the wind up them. However removal may result in more problems! Find a professional who will recommend the height at which they should be topped to minimise problems.
How you keep this away from being a known dispute is probably the biggest difficulty.
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