cut them all.
Hickory, elm, ash: cut, split into staves, pull the bark off while it's wet. Shellac the ends and back of the stave.
Locust and Osage: cut, split into staves, work the backs to one growth ring ASAP, shellac the ends and especially the back of the stave. Getting the bark and sapwood off osage will keep the wood grubs away but you'll have to shellac the back of the stave thoroughly to prevent checking.