Just yesterday I was having an email discussion with a friend about the origin of Where Have All the Flowers Gone melody, when lo and behold a song came on the radio from the other room, and I listened through the wall then went into the room and listened further. Was this some miraculous sign from some folk god in heaven? My wife, who had been listening to the radio (WFUV-FM Bronx NYC) in that other room quickly looked up the title on their playlist, which I loaded into my desktop. https://www.youtube.com/watch?v=fqrZSNN0_98 This particular iteration of The Banks of Sweet Dundee is a jig-like rendering that bears a striking resemblance to Pete's melody, especially if slowed down, as he said he did. It is not a perfect match, but does sound enough like Pete's melody that I thought to write out my story. It is often found to be the case that several songs can be derived "genealogically" from an older melody that has been around for centuries, this one being a Scots/Irish song that made it to the Catskills and was being knocked around in folk music circles that Pete frequented in the 1950's. May require a bit more research to establish provenance, but give a listen: https://www.youtube.com/watch?v=fqrZSNN0_98   
The Banks of Sweet Dundee
Mark Levy