Joe, your version is similar to the notation I have in John M Feierabend's The Book Of Lullabies . Some different notes, but essentially the same song.
Mama Lisa also has a couple of versions of it, although the studio recording by Cher and Gene Klosner has a very different tune.
I'm looking to teach it to my mommy-and-me class this year, so I've been looking for recordings, and for the correct person to attribute the lyrics to.
Thanks for all the info people have provided here.
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sj