You also have the problem of evidence.

What if the neighbor either (a) denies that he cut the branches; or (b) claims that your mother gave him permission to cut the branches?

If it comes to your mother's word against the neighbor's, I would say your mother already has a slight preponderance of evidence on her side. It's simply a lot easier to believe that the neighbor violated your mother's rights than that some unknown person cut the branches, or that your mother would give permission and later deny that she had done so (unless your mother is very forgetful).

But it would be good to have additional evidence.

Take pictures as soon as possible, showing the tree in its present state, the branches on the ground, and the area where the neighbor wants to build his addition, if it's visible from your mother's property. And if possible, find some pictures of the tree before the branches were cut. It would also help to have a written opinion from a tree expert about the value of the tree before and after the trimming, and what its prognosis is.

It would be great if you could somehow entice the neighbor to put some admission in writing. You, An Buachaill, are in a great position to do this. If you can somehow get the neighbor's email address, send him an email saying that your mother said that "something happened" to her tree but you don't quite understand what it was. Ask him if he can explain. Try not to tip him off that you are gathering evidence against him.

If you're lucky, your neighbor will readily admit what he did. He might even believe (erroneously) that he had a right to do it or that he was doing your mother a favor.

In the US, in small claims court, emails and even phone text messages are readily accepted as evidence, and many a defendant has been convicted by his own words.

Does Scotland have some equivalent to our small-claims court, where neither the plaintiff nor the defendant is allowed to bring a lawyer?