The Carter Family sang two close versions- "Bury Me Beneath the Willow," and "Bury Me Under the Weeping Willow." Neither is in the DT. Words under the first title in the DT are an abbreviated version; by (?)Guthrie or (?)Houston? Not the lyrics given by Sandburg, pp. 314-315 ("O Bury Me Beneath the Willow"), but a version of the same song.

There are steam-dozen songs about females buried beneath the willow.

The song in Belden, collected in 1909, is a version of the same song, but some other citations, could be to different songs with the same theme of burial beneath a willow.

Looking through 19th c. song sheet collections, among the females planted beneath the willow are Annie, Kate, Ella Lea (Lee), Cora Lee, Nell, Mary, Carrie Lea, and several anon. On the male side are mariners, soldiers, anon. etc.
Stephen Collins Foster wrote one of the most popular; "Under the Willow She's Sleeping."

Finding the one, true, 19th c. antecedent will not be easy.