This section is from the book "An Illustrated Flora Of The Northern United States, Canada And The British Possessions Vol1", by Nathaniel Lord Britton, Addison Brown. Also available from Amazon: An Illustrated Flora of the Northern United States, Canada and the British Possessions. 3 Volume Set..
Annual or perennial, often somewhat shrubby herbs, with terete, but usually striate erect or prostrate stems. Leaves alternate, leathery or somewhat fleshy, sometimes plicate, articulated to the ocreae. Ocreae lobed when young, at length lacerate, hyaline, not fringed. Inflorescence axillary, consisting of clusters bearing normally several flowers at each node throughout the plant or confined to the branches and branchlets. Sepals 5 or 6, mostly green with white or pink or yellow margins, 2 wholly interior, 2 wholly exterior and 1 with one edge exterior and one edge interior. Stamens varying from 3 to 8, often 5 or 6, included; filaments, at least the inner ones, dilated. Styles 3, usually distinct, sometimes very short. Achenes 3-angled, included or slightly exserted, brown or black, granular or smooth and shining. Endosperm horny. Cotyledons incumbent. [Greek, many-knees, from the swollen joints of some species]
About 100 species, of wide geographic distribution. Besides the following, some 10 others occur in the western parts of North America. Type species: Polygonum aviculare L.
* Stem and branches terete and usually striate.
tAchenes much exserted from the calyx.
Plants prostrate; achene broad.
Sepals decidedly petaloid, very broad, much overlapping and lax at maturity; achene acutish or
Sepals slightly petaloid, rather narrow, scarcely overlapping and appressed; achene acuminate.
Plants erect; achene narrow.
Flowers hidden in the ocreae; sepals about 1/2" long; achene slender.
Flowers exserted; sepals 1"-1 1/2" long; achene stout.
++Achenes included in the calyx, or exposed at the tip.
Sepals with white or pink margins.
Pedicels not exserted from the ocreae; sepals less than 1 1/4" long at maturity.
Achenes with striate faces.
Mature sepals over 3/4"long; achenes acute.
Mature sepals less than 3/4" long; achenes acuminate.
Achenes with granular or nearly smooth faces.
Plants prostrate; leaves broad; mature sepals about 3/4" long.
Plants erect or nearly so; leaves narrow; mature sepals about 1" long.
Pedicels exserted; sepals over 1 1/4" long at maturity.
Sepals with yellowish or greenish margins.
Leaves broad; achene mostly dull.
Leaves narrow; achene mostly shining.
Achenes much longer than wide; mature calyx tapering at the base.
Achenes about as wide as long; mature calyx auriculate-cordate at the base.
** Stem and branches angled.
Leaves plicate; fruiting pedicels short, erect.
Leaves flat, margins revolute; fruiting pedicels long, deflexed.