I also advise against self-tapping screws for this application . . . . they likely won't work, and may wind up breaking off inside the tree, giving you a project you didn't want in the first place . . .
Self-tappers are fine with sheet metal, not so fine for stuff like this.
Don't ask me how I know.
"Here is your country. Cherish these natural wonders, cherish the natural resources, cherish the history and romance as a sacred heritage, for your children and your children's children. Do not let selfish men or greedy interests skin your country of its beauty, its riches or its romance.” T.R.