branch – ветка, ветвь; отделение, филиал, отрасль

branch (n.) (из старофранцузского «branche» с теми же значениями); из позднелатинского «branca» - отпечаток, позже – коготь, клешня, лапа, рука; происхождение неизвестно, возможно из гэльского.
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 (cf. 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.) (ветвиться, отрастать, ответвляться)
"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.

1826:
BRANCH, s. the limb of a tree, an arm of the sea; F. “branche”; Arm. “brank”; βράχίων; L. “brachium”. Ветви дерева, морской залив (рукав).
См. «brake» (5).