I know its very low on the btu scale. 12.1 million BTUs per cord. It's worse than cottonwood btu wise, and box elder at 17.9 blows it away. Willow is like 14 mbtu's per cord.
I have heard of catalpa or catawba, but I have never burned it. It is used in guitar making as tonewood.
Wet wood of any species sucks. Tree of heaven stinks when you cut it, and is fairly lightweight once dry so not very many BTUs, but it burns and doesn't stink once it has dried out- at least the stuff I heated with last year didn't, but those particular TOH trees didn't stink so bad when I cut them either, so maybe there is some variability. It is invasive and I am trying to eradicate it. I sure as hell am not going to cut it down and NOT put it through the wood stove. Heat is heat. I just put up 2/3 of a cord of it a couple months ago and my barn smells like rotten peanut butter right now, but the ashes smell the same as any other and it is going out through the chimney. At least it splits easily. I cut trees that have either fallen down or need to come down for other reasons- I don't go looking for firewood trees. If I cut it, I burn it.
Well, if you have it there you may as well burn it. I burn birch that I cut on my property here. Not the best wood, and it absolutely has to be split or it will rot pretty fast. The bark is waterproof, so it traps the moisture of the wood when bucked up. No smells involved with that stuff though.
Birch burns nicely. It starts nice, makes good heat, but doesn't last terribly long. It's pretty farwood too.
Last year I cut a bunch of Basswood down around the vegie garden for better sunlight. About 6 cords, it all went into a wet spot for "tractor road". Tree of Haven go's directly to a large burn pile.