Here is the root of a tree-of-heaven that we cut down and dug up earlier this year. I decided that 6-8 months was long enough for it to sit around, and tossed it onto the fire today.
It turns out to be difficult to get a good picture of the fire with my camera-phone, but this is what that hunk of junk wood looked like a few minutes after I put it on the coals. I reckon we got around 30-40 minutes of usable heat out of it, or maybe as much as an hour before I had to go get more wood.
The quality of the wood we're burning at the moment is poor, and it produces a lot of ash that has to be dug out of the fireplace, but it is free, aside from our labor in obtaining it, so any non-zero heat content is a win.