Well, the identity of the Mystery Tree has been solved, I think. My friend, Gordon, called right after I posted to say it's a Shaving Brush Tree, Pachira fastuosa (according to Successful Gardening in The Magic Valley of Texas) or Bombax ellipticum (according to just about every other website).
Native to Mexico, this deciduous tree will grow to about 30 feet. I would have guessed that it is very tender but one website claims it is hardy to 18 degrees. Although it is coveted for it's flashy pink brush-like flowers, the leaves are also very ornamental. I will be watching for them this spring. It is said that the new leaves are a bright red turning light green as they age. Another bonus is that Shaving Brush Tree needs little water.
SamiG did suggest that it could be a Pink Powder Puff, Calliandra surinaemensis. I couldn't find much info on Pink Powder Puff but the Forest Service has a good fact sheet. They mention that Pink Powder Puff blooms throughout the year - and I don't think this one does. But I'll keep an eye on it and let you know if I've changed my mind.