Like Hillbilly said, the river birch peals off in thin layers. What you see curling on the tree, if you pull a piece of that bark off it is made up of 3-4 thin layers. The outer layer has the white-gray color, the inner layers are a reddish brown more solid color.
I was thinking of separating the layers and only using the outer layer, then gluing that down to the back and wrapping it like you would a snake skin until the glue drys. I'm not sure if it would need to be soaked first or not. I've got a bow in the works that I might try it on. Would that thin layer hold up to years of shooting when it's sealed?