@bullworth
The nub of this is that your tree is clearly irritating your neighbour considerably.
That is called nuisance.
You on the other hand feel that there has been unlawful trespass - and that is triable per se - in other words you do not have to prove damage.
If you want to bring this to head:
Issue a summons under the Petty Sessions (Ireland) Act 1851.
Information can be laid before a District Court Judge on an ex parte and can be done by you.
If you have all your facts then you will find out soon enough what the story is.
I would predict that the Judge might tell you that you have an arguable case but your neighbours potential defence would make the outcome of this unpredictable - as we have only your side to go on.
His tale might be such that many hints have been given to you to cut the branches back - but you are not listening (and some here might be sympathetic to your neighbour on that front). On the other hand he may have said nothing and made no attempt to contact you.
The judge will be a little bemsued as to why you have not attempted some sort of contact but if you are persistent he may let this go forward. I think there are in fact two different complaints here and you might er not win. A simple strategy is to cut the branches. A longer strategy is to communicate with your neighbours as you may need them in the future.