branch n.

early 15c., pedigrue, "genealogical table or chart," from Anglo-French pe de gru, a variant of Old French pied de gru "foot of a crane," from Latin pedem accusative of pes "foot" (from PIE root *ped- "foot") + gruem (nominative grus) "crane," cognate with Greek geranos, Old English cran; see crane (n.)).

On old manuscripts, "descent" was indicated by a forked sign resembling the branching lines of a genealogical chart; the sign also happened to look like a bird's footprint. On this theory the form was influenced in Middle English by association with degree. This explanation dates back to Skeat and Sweet in the late 1800s. The word obviously is of French origin, and pied de gru is the only Old French term answering to the earliest English forms, but this sense is not attested in Old French (Modern French pédigree is from English). Perhaps it was a fanciful extension developed in Anglo-French. Other explanations are considered untenable.

The crane was at the time in question very common in England and France, and it figures in many similes, proverbs, and allusions. The term appears to be extant in the surname Pettigrew, Pettygrew .... [Century Dictionary] 

Meaning "ancestral line" is mid-15c.; of animals, c. 1600. Related: Pedigreed.

Middle English bough, from Old English bog "shoulder, arm," extended in Old English to "twig, branch of a tree" (compare limb (n.1)), from Proto-Germanic *bogaz (source also of Old Norse bogr "shoulder," Old High German buog "upper part of the arm or leg," German Bug "shoulder, hock, joint"), from PIE root *bhagu- "arm" (source also of Sanskrit bahus "arm," Armenian bazuk, Greek pakhys "forearm"). The "limb of a tree" sense is peculiar to English.