One of the plants that reseeds quite a bit in the shade garden is Tovara Virginia or Painter's Palette. It brightens up the shadowy places beneath trees with it's colorful leaves. Late in the summer, its flowers rise on tall, bare stalks, looking more like tiny red beads than flowers. In 2001, David Barnett posted a dozen shade loving plants on the Arlington Organic Garden Club webpage. Barnett's list is here and includes Tovara, Hardy Begonia, Turk's Cap and others.
Missouri Botanical Garden's MOBOT Plantfinder page says the common name is knotweed. They also say it is a Missouri native they call Polygonum virginianum.
At the URI Master Gardener's Favorite Plants site, the plant is recommended as Polygonum virginianum or Tovara virginiana. Their term for her willingness to fill a bed is "an enthusiastic self-sower". Rhode Island isn't in our horticultural zone but their list of plants is similar. Lots of great information and links at the site.
At Dave's Garden, the members who grow a plant they call Knotweed and Tovara is Persicaria or Fleece Flower. I have that in the shade garden, too. It's not the same plant at all.
Alice Joyce wrote about it in the San Francisco Chronicle last year. She said, "This genus keeps taxonomists occupied, as evidenced by recent name changes and much confusion: Synonyms include polygonum and persicaria, but Tovara v. 'Variegata' should not be (although it sometimes is) mistaken for fallopia, a towering, thuggish plant with a running habit."
Many plants such as Tovara (or Knotweed or Persicaria or Polygonum or .....) are available at plant swaps and club sales. An online search yielded one EBay seller (3 plants for $9.00) and Plant Delights Nursery has a beautiful one from Tasmania called Tovara 'Brushstrokes' (Brushstrokes Fleece Flower) for $12.
Mine came from Tulsa Perennial Club's annual sale for $5. Do you have Tovara? Do you like it in your garden?