branch has 12 distinct synonyms spread across 6 of its 8 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
a division of some larger or more complex organization
“a branch of Congress”
a part of a forked or branching shape
“he broke off one of the branches”
a stream or river connected to a larger one
any projection that is thought to resemble a human arm
“the arm of the record player”
Synonyms, antonyms and the broader and narrower relations for branch come from Princeton University’s WordNet. Full sources and licences.