1. Stem and petioles with retrorse prickles; leaves hastate or sagittate (with acute basal lobes).

2. Leaves hastate, the basal lobes divergent; achenes 2-sided.

P. arifolia

2. Leaves sagittate, the basal lobes with tips parallel; achenes 3-sided.

P. sagittata

1. Stem and petioles without prickles; leaves various.

3. Inflorescence an elongate, remotely flowered spike-like raceme (flowers solitary at nodes), the main (terminal) one 15–36 (–47) cm long; styles 2, soon elongating and becoming firm and hardened, persistent, hooked at the tip; leaf blades (at least the larger ones) (4–) 4.3–9 (–10) cm broad.

P. virginiana

3. Inflorescences shorter, often densely flowered (flowers clustered at nodes); styles 1 or 3, very short, deciduous, not hooked; leaves less than 4 cm broad.

4. Inflorescences 1 (–2), terminal, thick and densely flowered; leaves in some forms floating and in some forms subcordate (and in some, neither).

P. amphibia

4. Inflorescences few to many, axillary as well as terminal (if some plants with only 1–2, these very slender, less than 5 mm thick, and elongate); leaves neither floating (may be submerged) nor subcordate.

5. Ocreae without bristles (very rarely with a few minute cilia).

6. Peduncle with glands sessile or (usually) none; inflorescences rarely as thick as 9 mm, usually nodding; outer tepals with veins becoming strong and forked with recurved anchor-like branches toward margin; achenes (1.3–) 1.5–2.2 mm broad.

P. lapathifolia

6. Peduncle with few to many distinctly stalked glands; inflorescences at least 9 mm thick, erect; outer tepals with veins becoming obscure toward margin; achenes 2.3–3.3 (–3.7) mm broad.

P. pensylvanica

5. Ocreae with at least a fringe of bristles on the margin (often with additional, usually appressed, hairs or bristles).

7. Peduncles and ocreae with spreading hairs.

8. Hairs, at least on peduncles, gland-tipped; achenes ca. 1.5–1.7 mm broad; principal leaves lanceolate, up to 2.5 cm broad; inflorescences rather loosely flowered, less than 1 cm thick.

P. careyi

8. Hairs not gland-tipped; achenes ca. 3–3.5 mm broad; principal leaves ovate, at least (2.5–) 5 cm broad; inflorescences densely flowered, usually at least 1 cm thick.

P. orientalis

7. Peduncles and ocreae with hairs appressed or none.

9. Tepals covered with glistening yellowish glandular dots (especially noticeable on dry specimens); plant sharply peppery to taste (reaction may be briefly delayed).

10. Achene with sides granular or cellular in appearance; ocreolae often swollen by included flowers or fruit; tepals usually rose-colored at tip.

P. hydropiper

10. Achene smooth and glossy; ocreolae not swollen; tepals white-tipped or entirely green.

11. Inflorescences ± interrupted; the lower ocreolae mostly not overlapping, the lowest three internodes 20–40 (–55 mm) long; lowest ocreolae with marginal cilia present, 0.5–1.8 mm long; widest leaf blades 0.5–2.5 (–3) cm wide.

P. punctata

11. Inflorescences densely flowered, essentially uninterrupted; the ocreolae overlapping, except sometimes the lowermost, lowest three internodes of the inflorescence usually less than ca. 18 mm long; lowest ocreolae with marginal cilia usually less than 0.7 mm long or absent; widest leaf blades more than 2.4 cm wide.

P. robustior

9. Tepals not gland-dotted; plant not peppery to taste.

12. Inflorescences (at least the main ones) over 3 cm long, slender and loosely flowered; styles 3 and achenes 3-sided; plants perennial (often with stoloniferous base rooting at nodes); peduncles and stems above glabrous to ± densely strigose.

P. hydropiperoides

12. Inflorescences (0.5–) 1–3 cm long, densely flowered; styles 2 and achenes 2-sided (a few or very rarely all 3-sided); plants annual, from a small taproot; peduncles and stems glabrous or at most sparsely strigose.

13. Bristles of ocreolae about equaling or exceeding perianth; achenes all 3-sided.

P. longiseta

13. Bristles of ocreolae, if present, much shorter than perianth; achenes usually 2-sided (rarely all 3-sided).

P. maculosa