I looked at your pictures. That tree is not a Paulownia Elongata. Their leaves are heart shaped but also lobed and over a foot across and hairy on both sides. Also, the Elongata is non-invasive. They do not send up shoots, you can not propagate them from root or seed. They are sterile and can only be propagated with cuttings. I don't think it is a Paulownia tormentosa either. Altho they are invasive, send up shoots and can be easily sprouted from seed or root cutting. They too have big heart shaped hairy leaves. But the leaves of your tree look pretty small. It may be any one of the other 20+ Paulownias that are invasive. I don't know about them all. Here are photos of the leaf of the Royal Empress-Paulownia Elongata and the Chinese Empress- Paulownia Tormentosa. You can see well the hair on the new leaves. Fast growing trees .com has a lot of wonderful information on the Paulownia. Also has a list of them all.
Eric Hanson wrote:Here are my pictures. I have had to trim up the branches in order to mow under them. On branch did come down in heavy winds this winter, you can see it on the left. This does not bother me in the slightest as I know that this area will grow back in this summer as existing branches aggressively grow to fill the void.