This comes from a collection of African-American folk rhymes collected nearly a century ago. It's in a dialect.

Redhead Woodpecker - African American Children's Songs - Historical African American - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Intro Image

Notes

Holler tree = hollow tree (especially in the Southern US).

From the same source:

Why The Woodpecker's Head is Red:

Bill Dillix say to dat woodpecker bird:
"W'at makes yo' topknot red?"
Says he: "I'se picked in de red-hot sun,
Till it's done burnt my head."

Thanks and Acknowledgements

This rhyme can be found in Negro Folk Rhymes, Wise and Otherwise, with a Study by Thomas W. Talley of Fisk University (1922).

Standard English version by Lisa Yannucci. Image composed by Lisa.