branch:  Branch comes via Old French branche from late Latin branca ‘paw’, but its ultimate origins are not known. In other Romance languages it retains more of its original Latin sense (Spanish branca ‘claw’, for example, and Romanian brinca ‘hand, paw’). The semantic connection between ‘limb of a tree’ and ‘appendage of a person or animal’ is fairly straightforward (compare BOUGH).
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," 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.