Source 'Plants for Man' by Richard W. Schery (an excellent read if you are interested in the origins and uses of cultivated plants), also the on-line source Waynes' Word (
http://waynesword.palomar.edu/ecoph31.htm). According to these, Amaranthus caudatus and A. hybridus were native to the Andes (although A. hybridus is more widespread). A. leucocarpus is native to Mexico. A. cruentus, which Wayne calls an Arizona species, Schery says originally came from Asia.
In the US, the genus Prosopis (mesquite) only ranges as far east as southern Kansas and eastern Texas.
I'm growing groundnuts here and am hoping that the recently introduced kudzu bug doesn't find it tasty, as it does soybeans, yard long beans, and pole beans.
Whitetail and mule deer are both native within their respective ranges, the only reason some people are saying that they are invasive is because their numbers are booming in the artificial absence of predators (wolf, puma, human), although the automobile has become a significant deer predator in many suburban areas.
By "native" I mean a plant that arrived by its natural animal, wind, or water distributors in prehistoric times, not via a deliberate or accidental human introduction. But there are some who define "native" by saying it means anything that was here before the start of the European invasion of the Americas, or some who say, just because it has been here all of their lives (like dandelions or plantains), it is "native".
Most of the traditional native American domesticated crops in the region covered by the US were domesticated elsewhere (mostly from tropical America) and arrived via trade routes in prehistoric times. As these better crops were introduced, many of the less tasty, tedious to process native edibles dropped out of use, became very minor food inputs, or were relegated to famine food. Example, acorns are a very abundant, easy to collect, edible crop, but their high tannin content makes them tedious to process. So when something better was introduced (say, pecans), pecans were planted and the acorns stopped being collected.
So as I stated above, if we had to limit our plant cultivation to only (by my definition of) native species, we'd have a VERY limited diet, as almost all of our crops are introduced species (with a few that are so well adapted to their new homes that they have become invasive). So those who want to close the door to any introduction of new plant species are also closing the door to the introduction of any potential new crops (although by now many of the world's potential crops have already been introduced).