Old English bog "shoulder, arm," extended in Old English to "twig, branch of a tree" (compare limb (n.1)), from Proto-Germanic *bogaz (source also of Old Norse bogr "shoulder," Old High German buog "upper part of the arm or leg," German Bug "shoulder, hock, joint"), from PIE root *bhagu- "arm" (source also of Sanskrit bahus "arm," Armenian bazuk, Greek pakhys "forearm"). The "limb of a tree" sense is peculiar to English.
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