bough (n.)

"part or member," Old English lim "limb of the body; any part of an animal body, distinct from the head and trunk;" main branch of a tree," from Proto-Germanic *limu- (source also of Old Norse limr "limb," lim "small branch of a tree"), a variant of *lithu- (source of Old English liþ, Old Frisian lith, Old Norse liðr, Gothic liþus "a limb;" and with prefix ga-, source of German Glied "limb, member").

The unetymological -b began to appear late 1500s for no etymological reason (perhaps by influence of limb (n.2)). The Old English plural was often limu; limen and other plural forms in -n lasted into Middle English. Since c. 1400 especially of a leg; in Victorian English this usage was somewhat euphemistic, "out of affected or prudish unwillingness to use the word leg" [Century Dictionary]. However in Old and Middle English, and until lately in dialects, it could mean "any visible body part":

The lymmes of generacion were shewed manyfestly. [Caxton, "The subtyl historyes and fables of Esope, Auyan, Alfonce, and Poge," 1484]

Hence, limb-lifter "fornicator" (1570s). Limb of the law was 18c. derisive slang for a lawyer or police officer. To go out on a limb in figurative sense "enter a risky situation" is from 1897. Alliterative life and limb in reference to the body inclusively is from c. 1200. Obsolete limb-meal (adv.) "limb-from-limb, piecemeal" is from late Old English lim-mælum.

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, said to be probably from Gaulish. The connecting notion would be the shape (compare pedigree).

In English it replaced native bough. The meaning "local office of a business" is recorded by 1817, from the earlier sense of "component part of a system" (1690s).