There are actually several different types of wisteria. One variety is native to the Southeastern United States. The others are imports from Asia. The Asian wisteria has all but taken over the native wisteria which is a less vigorous grower. Wisteria from large retailers, such as Wal-Mart and Home Depot is usually the Asian variety. Most of the time, the sales staff will not know that there is a difference.
Wisteria will die back in the winter, but it starts growing fairly early in the spring. It blooms around the end of April and through May. Its stems can root at just about any point. It roots can easily sprout leaves. When cut back it suckers and its flowers release thousands and thousands of seeds. When the conditions are favorable, it can grow as fast as kudzu. It will still grow even when the conditions are not so favorable.
Wisteria also puts out a distinctive smell when cut or pulled from the ground. This smell is produced to drive off less aggressive vegetation. Day lilies can crowd it out, but they have to be fairly thick to do so. Forsythia and periwinkle, both appear to be companion plants to wisteria. That is to say that neither takes over the other and they grow together.
The best way to get rid of wisteria, is to pull it up every time you can. The roots and stems will break off when you pull with sufficient force, but it ain't easy. Cut it back and it will sucker. The easiest time to pull it out is actually in the winter when it is mostly dormant. Do it before the ground gets hard and you can make huge inroads. To get it off your trees, cut the vine anywhere between the ground and the tree trunk in the spring just before It starts to bloom. Once the part on the tree trunk dies off, you can easily pull it off provided, it's not too twisted. But you can't slack off for a season. Pulling and cutting wisteria must be part of your routine to be successful at keeping it at bay.
If you currently have wisteria in your garden at home, remember that this plant can be aggressively pruned without ill effect. Prune it. Aggressively. It needs it. Also remember that it cannot be neglected and then expected to behave itself and not take over.