The "root" of the problem, I believe is that the downspout leads to the curb, and I think that the drainpipe has been invaded by roots from our 80 yr. old sugar maple. Even in the summer, the water just eked out of the drain, it never had a flow to it. Now that it's winter, I suspect that the still water has frozen from the drain pipe at the tree and has worked its was freezing backwards up through the downspout up to the third story gutter.
I've read about heat cables on the roof. Can you run them down through underground drain pipes as well? And is this the best course of action?