The issue of overhanging branches from a neighbours garden is not something to take lightly. There have been numerous cases when neighbourly relations have reached an all time low because of something so simple as a plant encroaching on another property. Your neighbour does have a right to cut the branches or shoots which grow over his legal boundary but he should return them to you afterwards.
A better option would be to discuss the issue outright with your neighbour and certainly before it gets to the point where he take sthe matter into his own hands. You may be able to reach an agreement which suits you both and avoids the issue escalating further.