Folks have made natural forks from mesquite and citrus. It is best to stay away from pine. I don't know about the others you mentioned. In my part of the country any tree that produces a fruit or nut is fair game.
Eucalyptus is a hard wood, so it should be ok.
Keep in mind that my uncle is shooting a tree fork so thin and soft, that a new born could snap, and he shoots heavy, hunting rubber. He just holds the frame right behind the attached rubber, so no strain goes to the body of the slingshot.
I did give him a nice Wasp Delta Wing though ????