Badger, thanks for your responce. Now, I know my plan is workable. I can see where 4x4 would be less prone to warping. But, I don't have access to a good bandsaw. I will be cutting them on a table saw, so I'm thinking 2x2 will be easier to work with. But I'll check the grain before I reduce them too much. It might be that I need 2x3 to have room to lay the grain out straighter down the limbs.
My friend, Brian, bought his sawmill 10 years ago so he could cut wood on his farm to build a cabin. So I'm pretty lucky, all it will take is a 5 hour drive round trip and probably a case of beer.
Oh, and I'm making a bow for him. Last fall I asked Brian if he had any ironwood and/or dogwood on his farms that I could cut. I told him about what I was looking for. Two weeks later he calls to tell me he has 2 ironwood logs and a dogwood log for me. So, he might even fall a tree for me. Gotta love friends.