Etymology
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*bhel- (3)
Proto-Indo-European root meaning "to thrive, bloom," possibly a variant of PIE root *bhel- (2) "to blow, swell."

It forms all or part of: blade; bleed; bless; blood; blow (v.2) "to bloom, blossom;" bloom (n.1) "blossom of a plant;" bloom (n.2) "rough mass of wrought iron;" blossom; cauliflower; chervil; cinquefoil; deflower; defoliation; effloresce; exfoliate; feuilleton; flora; floral; floret; florid; florin; florist; flour; flourish; flower; foil (n.) "very thin sheet of metal;" foliage; folio; folium; gillyflower; Phyllis; phyllo-; portfolio; trefoil.

It is the hypothetical source of/evidence for its existence is provided by: Greek phyllon "leaf;" Latin flos "flower," folio, folium "leaf;" Middle Irish blath, Welsh blawd "blossom, flower;" Gaelic bile "leaflet, blossom;" Old English blowan "to flower, bloom."
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*pete- 
*petə-, Proto-Indo-European root meaning "to spread."

It forms all or part of: compass; El Paso; expand; expanse; expansion; expansive; fathom; pace (n.); paella; pan (n.); pandiculation; pas; pass; passe; passim; passacaglia; passage; passenger; passport; paten; patent; patina; petal; spandrel; spawn.

It is the hypothetical source of/evidence for its existence is provided by: Greek petannynai "to spread out," petalon "a leaf," patane "plate, dish;" Old Norse faðmr "embrace, bosom," Old English fæðm "embrace, bosom, fathom," Old Saxon fathmos "the outstretched arms."
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*bhel- (2)
Proto-Indo-European root meaning "to blow, swell," "with derivatives referring to various round objects and to the notion of tumescent masculinity" [Watkins].

It forms all or part of: bale (n.) "large bundle or package of merchandise prepared for transportation;" baleen; ball (n.1) "round object, compact spherical body;" balloon; ballot; bawd; bold; bole; boll; bollocks; bollix; boulder; boulevard; bowl (n.) "round pot or cup;" bulk; bull (n.1) "bovine male animal;" bullock; bulwark; follicle; folly; fool; foosball; full (v.) "to tread or beat cloth to cleanse or thicken it;" ithyphallic; pall-mall; phallus.

It is the hypothetical source of/evidence for its existence is provided by: Greek phyllon "leaf," phallos "swollen penis;" Latin flos "flower," florere "to blossom, flourish," folium "leaf;" Old Prussian balsinis "cushion;" Old Norse belgr "bag, bellows;" Old English bolla "pot, cup, bowl;" Old Irish bolgaim "I swell," blath "blossom, flower," bolach "pimple," bolg "bag;" Breton bolc'h "flax pod;" Serbian buljiti "to stare, be bug-eyed;" Serbo-Croatian blazina "pillow."

An extended form of the root, *bhelgh- "to swell," forms all or part of: bellows; belly; bilge; billow; bolster; budget; bulge; Excalibur; Firbolgs.

An extended form of the root, *bhleu- "to swell, well up, overflow," forms all or part of: affluent; bloat; confluence; effluent; effluvium; efflux; fluctuate; fluent; fluid; flume; fluor; fluorescence; fluoride; fluoro-; flush (v.1) "spurt, rush out suddenly, flow with force;" fluvial; flux; influence; influenza; influx; mellifluous; phloem; reflux; superfluous.
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*pet- 

Also petə-, Proto-Indo-European root meaning "to rush, to fly." 

It forms all or part of: accipiter; appetence; appetite; apterous; apteryx; archaeopteryx; asymptote; centripetal; Coleoptera; compete; competent; eurypterid; feather; helicopter; hippopotamus; Hymenoptera; impetigo; impetuous; impetus; iopterous; Lepidoptera; ornithopter; panache; panne; pen (n.1) "writing implement;" pennon; peripeteia; perpetual; perpetuity; petition; petulance; petulant; pin; pinion; pinnacle; pinnate; pinniped; potamo-; potamology; propitiation; propitious; ptero-; pterodactyl; ptomaine; ptosis; repeat; symptom.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit pattram "wing, feather, leaf," patara- "flying, fleeting;" Hittite pittar "wing;" Greek piptein "to fall," potamos "river, rushing water," pteron, pteryx "feather, wing," ptilon "soft feathers, down, plume;" Latin petere "to attack, assail; seek, strive after; ask for, beg; demand, require," penna "feather, wing;" Old Norse fjöðr, Old English feðer "feather;" Old Church Slavonic pero "feather;" Old Welsh eterin "bird."

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