Having read a bit more - notably the Bow Construction FAQ I discovered that, of the woods commonly available near Flagstaff, oak, juniper, and cedar might be suitable. Checked with the Forest Service about getting a permit to cut a tree, and they said that cedar here is limited. In the end I went with oak because:
* the oaks here grow very straight.
* its the most common, and I might as well make my first mistakes on common wood.
* there's a stand of them growing in our front yard, which needs thinning anyways. (Hurrah for minimising effort.)
Jamie is leaning against our chosen vitim.
Following the advice from the FAQ, we picked a tree from 4-6 inches in diameter (well, 12-16 inches in circumference, which is easier to measure before you cut it down) and with a fairly constant width. With a little help from Jamie, I cut down the tree and stripped off the branches, and, since I couldn't find any references to whether you seasoned it before or after splitting, put it up in the rafters of the shed. This would have been about the beginning of April. (Since then I have found at least one reference to someone splitting first and then seasoning, but even there it was unclear as to how much of the shaping happened before the staves were put away to season. Hopefully having done it the other way round isn't going to cripple the attempt from the start. We'll see.)
The rope alongside is roughly the length of the finished bow. I cut the staves long so I could work around flaws or knots to a certain extent.