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Keyed in multiple places:

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Key to Polygonaceae

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1 Woody vine, climbing by tendrils; [subfamily Polygonoideae, tribe Coccolobeae].
  2 Leaf base deeply cordate
  2 Leaf base truncate to broadly cuneate
1 Herb (sometimes very robust and rather woody), herbaceous vine, tree or shrub (Coccoloba), or woody (Muehlenbeckia) or somewhat woody (Fallopia baldschuanica) vine lacking tendrils.
    3 Tree or large shrubs (> 2 m tall when mature); bark of medium to mature trees peeling.
      4 Leaves ovate to orbicular or sometimes lanceolate-elliptic (as in C. diversifolia); floral tube becoming fleshy, fruit not winged; [common in sandy coastal habitats, c. and s. FL]
      4 Leaves long elliptic to widely lanceolate; floral tube not becoming fleshy; fruit 3-winged [rare escape, disturbed areas in s. FL]
    3 Herbs, herbaceous vines, woody vines, or small shrubs (< 2 m tall); stem not exfoliating.
        5 Stem leaves (in our species) whorled; flowers in involucrate heads; ocreae absent; stamens 9; leaves densely white-tomentose on the lower surface; [of xeric situations of shale barrens and sandhills]; [subfamily Eriogonoideae, tribe Eriogoneae]
        5 Stem leaves alternate; flowers in various inflorescences (not involucrate); ocreae present; stamens (3-) 5-8 (-9); leaves glabrous or variously pubescent, but not densely white-tomentose; [of various habitats, including xeric ones]; [subfamily Polygonoideae].
          6 Tepals 6, in 2 series of 3 each; plants with leaves basally disposed, the largest basal (these withering in some species later in the season); [tribe Rumiceae].
             7 Tepals (the outer series) spinose
             7 Tepals foliose.
               8 Fruit 3-winged; basal leaves very large, 20-40 cm wide; inner and outer tepals similar; [plant cultivated, rarely persistent or escaped]
               8 Fruit 3-angled; basal leaves small to medium in size, 0.5-15 cm wide; inner tepals wider than the outer tepals; [plants common, mostly weedy]
                 9 Leaf blades of well-developed leaves (at least) hastate or sagittate; plants dioecious (rarely polygamo-monoecious), the flowers mostly unisexual; fresh foliage pleasantly acid to taste
                 9 Leaf blades not hastate or sagittate; plants synoecious (rarely with some dioecious or polygamo-monoecious individuals), the flowers normally bisexual (sometimes bisexual and unisexual flowers in the same inflorescence); fresh foliage “green” or bitter to taste
          6 Tepals mostly 5 in a single whorl; plants with leaves along the stem, lacking well-developed basal leaves.
                   10 Flowers in small clusters or very reduced racemes of 1-5 flowers, borne in the axils of normally sized or reduced leaves; plants erect or sprawling herbs with stems < 1 m long, from taproots; leaves jointed at base; [tribe Polygoneae].
                     11 Stems with 4 obscure ribs or angles (or lacking apparent ribs); leaf venation parallel, with inconspicuous secondary veins, longitudinally plicate; anthers pink-purple
                     11 Stems with 8-16 distinct ribs; leaf venation pinnate, the secondary veins apparent; anthers whitish yellow
                   10 Flowers in diffuse axillary panicles, or in terminal or long-peduncled axillary racemes, corymbs, or heads; plants various, either erect or sprawling herbs, or erect, robust, and suffrutescent herbs, or climbing herbaceous, suffrutescent, or woody vines, or suffrutescent bushy herbs; leaves not jointed at base (except Polygonella).
                       12 Leaves cuneate at the base, either linear, spatulate, or oblanceolate, mostly < 4 cm long and < 5 mm wide; leaves jointed at the base; pedicels jointed at the base; [tribe Polygoneae]
                       12 Leaves cuneate, cordate, or hastate at the base, either lanceolate or ovate, mostly > 5 cm long and > 8 mm wide; leaves not jointed at the base; pedicels not jointed at the base.
                          13 Inflorescence corymbiform, terminal; achenes strongly exserted at maturity; tepals almost free, horizontally spreading, white, 3-4 mm long; [erect annual, uncommonly cultivated and rarely persistent or escaped]; [tribe Persicarieae]
                          13 Inflorescence paniculate, racemiform, or headlike, terminal and axillary; achenes enclosed in the perianth at maturity; tepals fused for much of their length, ascending, pink, green, or white.
                            14 Plants wiry-stemmed, woody vines with small leaves (< 2 cm long); flowers unisexual, the tubes of the pistillate flowers becoming fleshy in fruit
                            14 Plants herbs or with thick, suffrutescent to woody, stems with larger leaves; flowers bisexual or unisexual (if unisexual, then the tubes of the pitillate flowers not becoming fleshy).
                              15 Outer tepals neither keeled nor winged at maturity; inflorescence of spikelike racemes, heads, or sparse, interrupted racemes; [tribe Persicarieae]
                              15 Outer tepals keeled or winged at maturity; inflorescence a compound panicle of racemes; [tribe Polygoneae].
                                16 Plants erect, robust (1-4 m tall), woody, the stems generally over 1 cm in diameter, hollow; perianth enlarging in fruit; stigma fimbriate
                                16 Plants climbing or sprawling, herbaceous to somewhat woody, the stems slender; perianth usually not enlarging in fruit; stigma capitate or peltate
                                  17 Ocreae smooth; perianth greenish to yellowish or pinkish; achene glossy or dull black; [mostly of lower elevations].
                                  17 Ocreae reflexed bristly at the base; perianth white; achene glossy black; [of high elevation openings and woodlands]

Key to Polygonum

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1 Flowers in terminal or long-peduncled axillary racemes; branches adnate to stems, therefore appearing to arise internodally; plants suffrutescent bushy herbs
1 Flowers in small clusters or very reduced racemes of 1-5 flowers, borne in the axils of normally sized or reduced leaves; branches not adnate to the stem, and thus arising at nodes; plants erect or sprawling herbs.
  2 Stems with 4 obscure ribs or angles (or lacking apparent ribs); leaf venation parallel, with inconspicuous secondary veins, longitudinally plicate; anthers pink-purple
  2 Stems with 8-16 distinct ribs; leaf venation pinnate, the secondary veins apparent; anthers whitish yellow; [section Polygonum].
    3 Perianth bottle-shaped, constricted above the achene.
      4 Tepals keeled; fruiting perianth divided about 1/3 of its length; stem leaves (1-) avg. 1.6 (-3)× as long as branch leaves
      4 Tepals not keeled; fruiting perianth divided about ¾ of its length; stem leaves (1.5-) avg. 2.5 (-4)× as long as branch leaves
    3 Perianth more open, not constricted above the achene.
        5 Inflorescences spikelike, the cymules borne in the axils of bracts shorter than the cymules
        5 Inflorescences axillary, the cymules borne in the axils of leaves longer than the cymules (though often shorter than primary leaves of the stem).
          6 Ocreae pruinose; mature tepals notably white or pink, loosely spreading; achenes smooth, glossy; [of maritime situations]
          6 Ocreae not pruinose; mature tepals yellow-green, white, pink, or reddish, appressed to the achene; achenes either smooth and glossy or textured and dull; [mainly of inland and disturbed situations]; [P. aviculare complex].
             7 Outer 3 tepals cucullate, in fruiting condition distinctly surpassing the inner 2 sepals and concealing them.
               8 Plants prostrate; leaves 2.5-5.6 (-10)× as long as wide
               8 Plants ascending to erect; leaves 4-12× as long as wide.
                 9 Plants bluish green when fresh, turning dark brown to black when dried; leaves rounded or obtuse at the apex; pedicels 1-2 mm long; stem leaves 1-2.5 (-3.5)× as long as the branch leaves
                 9 Plants yellowish green when fresh, not darkening when dried; leaves acute to acuminate at the apex; pedicels 2.5-6 mm long; stem leaves 2.1-3.5 (-4.2)× as long as the branch leaves
             7 Outer 3 tepals flat, equaling or shorter than the inner 2 sepals.
                   10 Perianth tube 40-57% of the perianth length.
                     11 Tepals green, margins pink or red (rarely white), with branched veins; plants prostrate to ascending, with 3-15 stems; leaf blades 2.8-5.7 (-6.5)× as long as wide
                     11 Tepals green or reddish brown, margins white, with unbranched veins; plants usually ascending, with 1-7 stems; leaf blades (3.40) 4.2-9.2× as long as wide
                   10 Perianth tube 15-40% of the perianth length.
                       12 Leaf blades 2-4.5× as long as wide; perianth (2.3-) 2.8-4.7 (-5) mm long; achenes (of the early season) (2.1-) 2.7-3.7 mm long
                       12 Leaf blades (3.4-) 4.2-9.2× as long as wide; perianth 1.9-3.4 mm long; achenes (of the early season) 1.2-1.8 mm long