Branch
Middle English: from Old French branche, from late Latin branca ‘paw’.
wiktionary
From Middle English branche, braunche, bronche, from Old French branche, branke, from Late Latin branca(“footprint”, later also “paw, claw”) (whence Middle High German pranke, German Pranke(“paw”)), of unknown origin.
Perhaps of Celtic origin, from a hypothetical Gaulish *vranca, from Proto-Indo-European*wrónk-eh₂. [1] If so, then Indo-European cognates include Old Norse vró(“angle, corner”), Lithuanian rankà(“hand”), Old Church Slavonic рѫка(rǫka, “hand”), Albanian rangë(“yardwork”).
The verb is from Middle English braunchen, from the noun.
etymonline
branch (n.)
c. 1300, braunch, "division or subdivision of the stem of a tree or bush" (also used of things resembling a branch in its relation to a trunk, such as geographic features, lines of family descent), from Old French branche "branch, bough, twig; branch of a family" (12c.), from Late Latin branca "footprint," later "a claw, paw," which is of unknown origin, probably from Gaulish. The connecting notion would be the shape (compare pedigree). Replaced native bough. Meaning "local office of a business" is first recorded 1817, from earlier sense of "component part of a system" (1690s).
branch (v.)
"send out shoots or new limbs," late 14c., also, of blood vessels, family trees, etc., "to be forked," from branch (n.). Meaning "to spread out from a center, radiate" is from c. 1400. Related: Branched; branching.
