Etymology
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willow (n.)
Old English welig "willow," from Proto-Germanic *wel- (source also of Old Saxon wilgia, Middle Dutch wilghe, Dutch wilg), probably from PIE root *wel- (3) "to turn, revolve," with derivatives referring to curved, enclosing objects. The change in form to -ow (14c.) paralleled that of bellow and fellow. The more typical Germanic word for the tree is represented by withy.
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pussy-willow (n.)

popular name of a type of common American shrub or small tree, by 1835, a country or children's word, from pussy (n.1) + willow. So called for the small and very silky catkins it produces in early spring. Also sometimes pussy-cat (1850).

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willowy (adj.)
"flexible and graceful," 1791, from willow + -y (2). Earlier "bordered or shaded by willows" (1751). Willowish is older (1650s) but only in reference to the color of willow leaves. Related: Willowiness.
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*wel- (3)
Proto-Indo-European root meaning "to turn, revolve," with derivatives referring to curved, enclosing objects.

It forms all or part of: archivolt; circumvolve; convoluted; convolution; devolve; elytra; evolution; evolve; Helicon; helicopter; helix; helminth; lorimer; ileus; involve; revolt; revolution; revolve; valve; vault (v.1) "jump or leap over;" vault (n.1) "arched roof or ceiling;" volte-face; voluble; volume; voluminous; volute; volvox; volvulus; vulva; wale; walk; wallet; wallow; waltz; well (v.) "to spring, rise, gush;" welter; whelk; willow.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit valate "turns round," ulvam "womb, vulva;" Lithuanian valtis "twine, net," vilnis "wave," apvalus "round;" Old Church Slavonic valiti "roll, welter," vlŭna "wave;" Greek eluein "to roll round, wind, enwrap," eilein "twist, turn, squeeze; revolve, rotate," helix "spiral object;" Latin volvere "to turn, twist;" Gothic walwjan "to roll;" Old English wealwian "roll," weoloc "whelk, spiral-shelled mollusk;" Old High German walzan "to roll, waltz;" Old Irish fulumain "rolling;" Welsh olwyn "wheel."
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osier (n.)

species of willow with tough, flexible branches used in basket-work, c. 1300, "a willow switch," from 14c. of the tree itself, from Old French osier, ozier "willow twig" (13c.) and directly from Medieval Latin osera, osiera "willow," ausaria "willow bed," a word of unknown origin, perhaps from Gaulish. Old English had the word as oser, from Medieval Latin.

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withy (n.)

Old English wiðig "willow, willow twig," from Proto-Germanic *with- "willow" (source also of Old Norse viðir, Danish vidje, Swedish vide, Old High German wida, German Weide "willow"), from PIE root *wei-  "to bend, twist" (source also of Avestan vaeiti- "osier," Greek itea "willow," Latin vītis "vine," Lithuanian vytis "willow twig," Polish witwa, Welsh gwden "willow," Russian vitvina "branch, bough").

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sallow (n.)

type of tall, shrubby willow plant of the Old World, Middle English saloue, from Old English sealh (Anglian salh), from Proto-Germanic *salhjon (source also of Old Norse selja, Old High German salaha, and the first element in the German compound Salweide).

This is reconstructed to be from PIE *sal(i)k- "willow" (source also of Latin salix "willow" (taken in botany as the genus name), Middle Irish sail, Welsh helygen, Breton halegen "willow"). French saule "willow" is from Frankish salha, from the Germanic root. It was used in Palm Sunday processions and decorations in England before the importing of real palm leaves began.

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catkin (n.)

"spike of a flowering tree or shrub (especially a willow or birch) after fruiting," 1570s, from Dutch katteken "flowering stem of willow, birch, hazel, etc.," literally "kitten," diminutive of katte "cat" (see cat (n.)). So called for their soft, furry appearance.

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*wei- 

also weiə-, Proto-Indo-European root meaning "to turn, twist, bend," with derivatives referring to suppleness or binding. 

It forms all or part of: ferrule; garland; iridescence; iridescent; iris; iridium; vise; viticulture; wire; withe; withy.

It is the hypothetical source of/evidence for its existence is provided by: Avestan vaeiti- "osier;" Greek itea "willow," iris "rainbow;" Latin viere "to bend, twist," vitis "vine;" Lithuanian vytis "willow twig;" Old Irish fiar, Welsh gwyr "bent, crooked;" Polish witwa, Welsh gwden "willow," Russian vitvina "branch, bough;" Old English wir "metal drawn out into a fine thread." 

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Avis 

U.S. car rental company, according to company history founded 1946 at Willow Run Airport in Detroit by U.S. businessman Warren Avis and named for him.

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