From what I've read, there are some areas where the Forest Service/Park Service are allowed to use power tools for maintenance activities. This seems like the exception rather than rule, however.

I have a friend who does trail maintenance in the Dolly Sodds and surrounding areas, and he has gotten quite adept at using a crosscut saw with his partner. I've seen it in action, and the efficiency of that tool, in the hands of skilled users, is quite impressive.