None of those however, told me what the offering should be, or how to ask for permission. Or what to do with the branch afterwards. So, can someone point me to a guide, or give an account of their own methods?
Standard practice in the group I work with is to leave a small offering that we feel is appropriate: if we're not sure, our standards are copper pennies or a few strands of hair. Several of us keep small stones or biodegradable substances handy for offerings. (I keep undyed spun wool, when I remember.) Some people use tobacco, which has a long tradition in many Native American traditions (and which is therefore quite appropriate if you live in those areas) but finding unpolluted tobacco is not necessarily simple if you don't smoke.
As far as what to do with it: depends. Both of mine had the bark stripped, and some light sanding, and that's it. I know people who do far more elaborate things (wire wrapping, adding stones, etc.) You can use glue that will hold stones or glass, or leather wrapping, too. Me, I go for simple. You can also paint them, or use ink or other coloring agents. (Just be careful that whatever you use will hold up to use, and not come off on your fingers!)