"Bury Me beneath the Willow" is an American folk song of unknown origin, included in several early 20th century collections. It is the story of an abandoned lover contemplating death — of sorrow? by suicide? — usually (but not always) sung by a woman. The Carter Family and the Delmore Brothers both recorded it in the 1930s, as did other early Bluegrass groups. I thought I remembered a version by Pete Seeger, or maybe the Weavers, but I cannot find it. Instead, I've included a similar song — with a theme of abandonment and death — at the bottom of this post.
Woody Guthrie with Sonny Terry. I think Cisco Houston is playing mandolin and singing on the chorus, but I can't be sure: