Ask Mr. Smarty Plants

Before we answer your question, let us tell you what we know about the tree. It is variously called "Royal Empress," "Royal paulownia" and "Princess Tree." It is called a lot of other things by people who either inherited this tree in their yards, bought and planted one, or had one planted by seedlings. However, this is a family website and we can't repeat those things. 

Begin by reading, from the Plant Conservation Alliance's Alien Plant Working Group Least Wanted : Royal Paulownia - Princess Tree - Royal Empress Tree - Paulownia tomentosa.

We would also like for you to read the comments on this Dave's Garden forum site on Paulownia tomentosa, in particular, the 7 negative comments.  Notice the comments about it showing up in disturbed ground, like where a swimming pool has been put in, and about the roots breaking up foundations, just from a seed that caught on the corner of that foundation and grew into a tree. Considering this tree is purported to grow 8 ft. a year, and that most trees have root system circumferences of 2 to 3 times the height of a tree, we would say your pool, sidewalks, house foundation and maybe the next-door neighbor's foundations are all at risk. If you planted it only a couple years ago, maybe it has not yet started seeding. If you let it live long enough to seed, not only will you be fighting a grove of these plants, but your pool will be a mess all the time. 

Our advice is to get rid of it while you still can. Roots will send up suckers trying to survive, and you will have to be after them all the time, and we just hope it has cast no seeds yet. Please, not in Austin, we are already having enough trouble with invasives.