I think in the interests of good neighbourly relations, you should pop round and advise that you are 'thinking of pruning the parts of his trees on your side ...'.
This takes any confrontation out of the situation - yes you are entitled to prune, but why not give the impression that you are checking it's OK first? It makes your neighbour feel good, and he is hardly likely to refuse, but will think well of you for having the thoughtfulness to ask first.
Of course, you get rid of the cuttings.