This is Chase072.
Motifs J2066.3 Men (animals) wait in vain for nuts to fall from tree.
Tuggle has a Creek myth in Swanton: rabbit chose sycamore and his tree "and nearly starved waiting for sycamore balls to drop. Since then he has stolen fruit from other animals."
Bear mentions K1035 stone (hard fruit) thrown into greedy dupe's mouth as related motif, also African, but it doesn't seem especially relevant; this is clearly a fool's errand type of story.
See also Dundes.
Baer: "This would appear to be another example of an African tale adopted by the American Indian."