A bough is a large branch from a tree. You know: “When the bough breaks, the cradle will fall...” A “lullaby” about a baby careening to the ground from a broken branch? (Nice.)
The original “Hush-a-bye” rhyme, some scholars say, was modified in America to reflect pilgrim children’s observations of Native American mothers hanging cradles from tree branches so that the wind would rock their babies to sleep. Why do poets use bough when branch and limb mean the same thing? Well, maybe because more words rhyme with bough than with those others. Also, “holly boughs” and “blossomy boughs” sound much more musical than “branches.” What rhymes with branches? Cattle ranches?