The full refain is
"To thee, to thee, derry derry to thee,
To thee, to thee, derry derry to thee,
And this was the burthen of his Song,
if thou wilt pipe Lad, I'll dance to thee."

Tune, as shown in Northumbrian Minstrelsy -

https://www.youtube.com/watch?v=gLG9w--0tMU&feature=youtu.be

I think the refrain is close cousin to Katie Bairdie (given the C17 practice of shoe-horning almost any words into almost any tune. But then, the whole tune seems to me a close cousin of the Arkansas Traveller as well, so you should not, perhaps, take my opinion that seriously.