I've built some half dz. bows from silver birch (this is called empirical evidence, fact-hungry enginerds). Depending on the density of the particular stave, silver birch ranges from über-crap to servicable bow wood. Crap in that a two-inch wide, perfectly flat belly with plenty of length per draw chrysals all over the place, while the densest staves will make a medium-set 40-45# flatbows at around 2" limb width.
American birches are as a rule denser and stronger (appreciably so) than silver birch. Dense and strong birches are found in Asia, too. Some of these have been planted extensively.
Tuukka