Real Dictionary

branch
A branch of an organization such as the government or the police force is a department that has a particular function.
countable noun: A branch of a business or other organization is one of the offices, shops, or groups which belong to it and which are located in different places.
countable noun: A branch of a subject is a part or type of it.
countable noun: The branches of a tree are the parts that grow out from its trunk and have leaves, flowers, or fruit growing on them.

branch in American English
any woody extension growing from the trunk or main stem, or from a main limb, of a tree or shrub
intransitive verb: to put forth branches; spread in branches
noun: a division or subdivision of the stem or axis of a tree, shrub, or other plant
transitive verb: to divide into branches or sections

branch in British English
noun: a secondary woody stem arising from the trunk or bough of a tree or the main stem of a shrub
verb: (of a tree or other plant) to produce or possess branches

branch- in British English
(in zoology) indicating gills