[*] 1. Heads radiate (obscurely so in n. 41, sometimes discoid in n. 36), the ray pistillate and fertile, the disk perfect but sterile. [+] Achenes turgid, triangular-obovoid; pappus none. 36. Polymnia. Involucral scales in 2 rows, the 5 outer leaf-like, the inner small. [+][+] Achenes flattened dorsally (obcompressed). 37. Silphium. Achenes wing-margined, in several rows; pappus none or 2 teeth. Scales thick, in several rows. 38. Berlandiera. Achenes wingless, 5--12 in one row, without pappus. Inner involucral scales obovate, outer smaller and more foliaceous. Western. 39. Chrysogonum. Achenes wingless, about 5; pappus a one-sided 2--3-toothed crown. Inner scales 5, chaff-like, the 5 outer longer and leaf-like. 40. Engelmannia. Achenes wingless, 8--10; pappus a scarious hispid crown. Outer scales (about 10) leaf-like, inner coriaceous with green tips. Western. 41. Parthenium. Rays 5, very short, persistent. Pappus of 2 small scales. Involucral scales short, roundish, in 2 rows. [*] 2. Fertile flowers 1--5, the corolla none or reduced to a tube; staminate corolla funnel-form. Pappus none. [+] Heads with 1--5 pistillate flowers. Receptacle chaffy. 42. Iva. Achenes short, thick. Involucre of few roundish scales. [+][+] Heads of two sorts on the same plant, the upper staminate with an open cup-shaped involucre, the lower pistillate, of 1--4 flowers in a closed bur-like involucre. 43. Ambrosia. Scales of staminate involucre united. Fruit 1-seeded. 44. Xanthium. Scales of staminate involucre distinct. Fruit 1--4-celled, 1--4-beaked. [*] 3. Heads radiate, or rarely discoid; disk-flowers all perfect and fertile. Anthers blackish. Pappus none, or a crown or cup, or of one or two chaffy awns, never capillary, nor of several uniform chaffy scales.--Leaves more commonly opposite. [+] Involucre double; the outer forming a cup. 45. Tetragonotheca. Outer involucre 4-leaved. Achenes obovoid; pappus none. [+][+] Involucre of one or more rows of separate scales. [++] Chaff of the flat receptacle bristle-shaped. 46. Eclipta. Ray short. Involucral scales 10--12, in two rows, herbaceous. [++][++] Chaff scale-like, embracing or subtending the achenes. [=] Receptacle high, conical or columnar in fruit. Pappus none or a short crown. 47. Heliopsis. Rays fertile. Achenes 4-sided. Leaves opposite. 48. Echinacea. Rays rose-colored, pistillate, sterile. Achenes short, 4-sided. Chaff spinescent. 49. Rudbeckia. Rays neutral. Achenes 4-sided, flat at the top, marginless. 50. Lepachys. Rays few, neutral. Achenes flattened laterally and margined. [=][=] Receptacle flat to convex. Achenes not winged nor very flat. 51. Borrichia. Achenes 3--4-angled; pappus a short 4-toothed crown. Shrubby. 52. Helianthus. Achenes flattened, bearing 2 very deciduous chaffy pointed scales. [=][=][=] Receptacle convex (rarely conical). Achenes flat-compressed laterally, winged or wingless, 2-awned. Leaves decurrent.