Oddly enough, there recently was a case almost exactly like this on "The People's Court" a popular daytime TV show in America. (Or maybe it was "Judge Judy." I sometimes watch them both!)

I believe the law on this in the US goes back to English Common Law and is probably the same everywhere.

The neighbor has the right to cut whatever branches overhang his property. He doesn't need to ask permission (but a polite person would at least notify you of his intentions and show that he understands the law). He then has the responsibility to dispose of the cut branches, or at least keep them on his own side. He does not have the right to cut more than this.

In the case on TV, the judge ordered the defendant to pay for replacement trees. However, the plaintiff didn't get as much money as she asked for; the judge decided the trees weren't worth as much as the plaintiff claimed.

As I recall, a mountain ash is a relatively short-lived tree to begin with. I used to have a couple.