Mr. Parnell,
The Shawnee were close confederates of the Ottawa, Chippewa and Potowattomii, which doesn't say crap about the wood. I have talked with Tribal Elders here abouts though and can tell you that "traditionally" The Ottawa preferred white Ash and only "occasionally the red cedar". While the Chippewa said that tradionally their preference was for ironwood and only rarely hickory.
I believe that if you use white ash, ironwood (hop-hornbeam) or any of the several hickories you wont be very far off. I know that Algonquian speaking nations clear out to Idaho (Bunji) used white ash if they could get it, or hickory if they could not.
Hope that helps you some. You may want to remember that nothing with Native Americans was wrote in stone, meaning that if a presentation bow was made for a warrior (for example) it could be made out of any "special" and costly wood the maker could get. Meaning there is a possibility it could be any white wood at all, if it was a special bow.