If I were asked to nominate at tree for the "Kudzu of Trees", it would be Ailanthus altissima, the Tree of Heaven. All parts of the tree were used in Chinese medicine, for conditions including mental illness, asthma, dysentery and epilepsy. It is still used in traditional Chinese medical practice.
It was brought to the US from China in 1784 to provide food for silk worms as it had in China. Like many invasive species, it seemed like a good idea at the time. The silk proved inferior and the Ailanthus silkmoth, while attractive, was not easily domesticated.Early on, it was noted to be easily propagated and would "grow anywhere". As a rule of thumb, when the words "exotic" and "easily grown and grows anywhere" occur in the same sentence, you are describing an invasive species looking for a place to happen.
Over time its aggressive growth and suckering habits became known. It was the subject of the book "A Tree Grows in Brooklyn", serving as a metaphor for the ability to thrive in a difficult environment. Its prolific growth in the cracks of urban sidewalks was the basis of its nickname, the "ghetto palm".
It has alternate, pinnately compound leaves with 11-41 leaflets. At first glance Ailanthus can easily be mistaken for a walnut, butternut, or sumac. These trees however have leaf margins with small teeth (except for winged sumac), while those of Ailanthus are smooth. The leaflets also have a distinctive notch or ear at the base with a gland on the underside. The leaves and stem have a bad odor when bruised, especially when bruised, giving rise to the name "stink tree".It is one of the fastest growing tree in the US, capable of growing 3 to 6 feet a year in the first four years of life. It grows in almost any disturbed soil as an early succession plant but doesn't compete well in established forests. Just a few trees along open areas can spread rapidly into open spaces and fields.
One secret of its success in reproducing lies in its roots. These sucker freely, spreading out a few feet and then shooting up as new trees. It also produces large crops of winged seeds which can spread over wide areas. Another possible factor is ailanthone, an allelopathic chemical which is produces by the Ailanthus roots. It is a very effective herbicide in laboratory tests, suppressing growth of many plants for weeks when placed in sterile soil. However, its real life effectiveness has not been scientifically proven.
There is no question however about the ability of Ailanthus to out-compete most plants. We have found it out in the middle of our 20 acre pasture and even a large collection in the backyard of our beloved MN vice-president. We need to eliminate it any where we can, but like kudzu, it is probably here to stay.
More information is at Wikipedia.
For methods of elimination see this PCA website.
A full discussion ailanthone is in this PDF.