"Now the fur upon the rabbit
Has the warm side fur side outside..."

From this verse forward, in the Milton Rosen/Smothers Brothers version, the words are sung to the tune of:

"O-le, O-le-an-na,
O-le, O-le-an-na...."

No wonder they like this song around the Cub Scout campfires: an old familiar tune, whether they recognize it or not.

Before that verse, however, there is no trace of the Oleanna tune; particularly not when cribbing "On the shores of Gitchee Goomee" from Longfellow.   For the opening verses the music is a sort of march, and I don't know where the tune comes from. Goes into double time for "Oh, he hunted over woods..." and so to "Oleanna."