branch (n.) Look up branch at Dictionary.com
c.1300, braunch, "limb of a tree" (also used of things analogous to it, especially geographic features), from Old French branche "branch, bough, twig; branch of a family" (12c.), from Late Latin branca "footprint," later "a claw, paw," 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.) Look up branch at Dictionary.com
"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.