Etymology
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arctic (adj.)
late 14c., artik, in reference to the north pole of the heavens, from Old French artique and directly from Medieval Latin articus, from Latin arcticus, from Greek arktikos "of the north," literally "of the (constellation) Bear," from arktos "bear; Ursa Major; the region of the north," the Bear being the best-known northern circumpolar constellation.

This is from *rkto-, the usual Indo-European root for "bear" (source also of Avestan aresho, Armenian arj, Albanian ari, Latin ursus, Welsh arth); see bear (n.) for speculation on why Germanic lost the word.

The -c- was restored from 1550s. From early 15c. as "northern;" from 1660s as "cold, frigid." As a noun, with capital A-, "the northern polar regions," from 1560s.
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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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Arctic Circle 

1550s in astronomy, in reference to a celestial circle, a line around the sky which, in any location, bounds the stars which are ever-visible from that latitude (in the Northern Hemisphere its center point is the celestial north pole); the concept goes back to the ancient Greeks, for whom this set of constellations included most prominently the two bears (arktoi), hence the name for the circle (see arctic). In Middle English it was the north cercle (late 14c.).

In geography, from 1620s as "the circle roughly 66 degrees 32 minutes north of the equator" (based on obliquity of the ecliptic of 23 degrees 28 minutes), marking the southern extremity of the polar day, when the sun at least theoretically passes the north point without setting on at least one summer day and does not rise on at least one winter one.

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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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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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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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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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