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).
"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.