Hand in hand invaders
The bright red leaves on the left are the late fall colors of the the Winged Burning Bush, Euonymus alatus. Sold in garden stores because it does make quite a display in the fall, sometimes appearing really bright red.
However, my observations of it confirm that it is one of those ambitious plants, prolific in producing seedlings nearby, but all over the place from bird transport of seeds. The National Park Service considers it a threat, http://www.nps.gov/plants/alien/pubs/midatlantic/eual.htm although Missouri doesn't yet. I have been removing the smaller ones when I find them; they are easy to identify with their green stems that have brownish wings along them lengthwise. And they pull up by the roots fairly easily.
These pretty red leaved shrubs are not that much of a threat unless they get loose and the young ones are not removed. However, the pretty green and yellow leaves on the right is from what I call a Bush honeysuckle, Lonicera maackii (Overgrowing Choke-plant or Amur honeysuckle), which has made the most invasive list in just about every site I visit. And there is no doubt in my mind that it does not play well in the forest garden. Fortunately the young ones do pull up very easily when the soil is wet enough. Identification is easy due to the same thing that makes it so invasive -- the leaves turn green and fill out weeks before anything else at that level, and stay green much later, in this case nearly a month after everything else (except the fire bush) has dropped theirs.
The big ones require chainsaw work. Once it is down to a very low stump, one can drip some strong poison directly on the freshly cut (i.e. no more than a couple minutes after making the cut). Needless to say, avoid dripping poison anywhere else.
Alternatively, covering the stump with something totally lightproof will also work, but that will take a lot longer to completely kill it. I use a pair of gallon pots, one slid into the other and rotated so the drainage holes do not line up and allow light in. The jury is still out on how well that will work, I'll let you know in a couple years.