From Elvis and Gladys by Elaine Dundy (Jackson: Univ. Press of Mississippi, 2004), page 132:

"...on Elvis' last day in school, at the children's request, the teacher gave him thirty minutes to perform before the class. The last song he sang for them was the traditional tune, "A Leaf on a Tree" ("the leaf on the branch, the branch on the tree, the tree in a hole, the hole in the ground, and the green grass grew all around, all around, etc.").

Assuming that's the right song, it's in the DT under the title GREEN GRASS GREW ALL AROUND.