From the song notes by Charles K. Wolfe in the book included with the Bear Family Box Set:

Bury Me Under The Weeping Willow was a song both Sara and Maybelle had known since childhood. Like many mountain songs, it was probably based on a 19th century parlor song though nobody has yet been able to identify an exact source.

(lists versions with a few comments)

One stanza which the Burnett-Rutherford version added at the end, but which the Carters did not use, was:

Upon my grave you'll plant a rosie,
Below my tomb a turtle dove,
To show the world I died to save him,
But I could not win in love.