I've explained in this thread (or another; there are several on the subject) that the tune Andy Irvine used for Willy of Winsbury really belonged to a completely different song: Fause Foudrage. It is a Scottish tune; he inadvertently copied it from the wrong page in Child's English and Scottish Popular Ballads. It doesn't belong, traditionally, to Willy at all.

I don't for a minute believe Paddy Graber's claim that he learned Arbutus from his mother; and especially not to that tune. I'm pretty sure that he made it up all by himself; but it appears that he won't say, which is a pity; it isn't bad work.

The other Arbutus referred to was written by Alfred Percival Graves, to a traditional Irish melody; but that, too, has been gone into in detail elsewhere, and the information can easily be found via the onsite search engine.