Branches

a division or subdivision of the stem or axis of a tree, shrub, or other plant.
a limb, offshoot, or ramification of any main stem:
the branches of a deer’s antlers.
any member or part of a body or system; a section or subdivision:
the various branches of learning.
a local operating division of a business, library, or the like.
a line of family descent stemming from a particular ancestor, as distinguished from some other line or lines from the same stock; a division of a family.
a tributary stream or any stream that is not a large river or a bayou.
Chiefly South Midland and Southern U.S. branch water (def 2).
Linguistics. (in the classification of related languages within a family) a category of a lower order than a subfamily and of a higher order than a subbranch or a group, as the Germanic branch of Indo-European.
Compare group (def 4a).
Computers. a point in a computer program where the computer selects one of two or more instructions to execute, according to some criterion.
Nautical. a warrant or license permitting a pilot to navigate in certain waters.
to put forth branches; spread in branches.
to divide into separate parts or subdivisions; diverge:
The main road branches off to the left.
to expand or extend, as business activities:
The bank has plans to branch throughout the state.
to divide into branches or sections.
to adorn with needlework; decorate with embroidery, as in textile fabrics.
branch out, to expand or extend, as business activities, pursuits, interests, etc.:
The business is branching out into computers.
Contemporary Examples

Confessions of an Obama Volunteer Tom Shone November 3, 2008
Can Gorillas Save the Democratic Republic of the Congo? Nina Strochlic April 27, 2014
Can Libraries Survive in an Era of Budget Cutbacks? Miranda Green February 15, 2013
A Justice Reflects Walter Isaacson June 28, 2009
Fresh Picks Nina Simonds April 19, 2011

Historical Examples

American Forest Trees Henry H. Gibson
The Leopard Woman Stewart Edward White
Painted Veils James Huneker
Buried Cities: Pompeii, Olympia, Mycenae Jennie Hall
The Bristol Royal Mail R. C. Tombs

noun
a secondary woody stem arising from the trunk or bough of a tree or the main stem of a shrub
a subdivision of the stem or root of any other plant
an offshoot or secondary part: a branch of a deer’s antlers

a subdivision or subsidiary section of something larger or more complex: branches of learning, branch of the family
(as modifier): a branch office

(US) any small stream
(maths) a section of a curve separated from the rest of the curve by discontinuities or special points
(computing) Also called jump. a departure from the normal sequence of programmed instructions into a separate program area
an alternative route in an atomic or nuclear decay series
verb
(intransitive) (of a tree or other plant) to produce or possess branches
(intransitive) usually foll by from. (of stems, roots, etc) to grow and diverge (from another part)
to divide or be divided into subsidiaries or offshoots
(intransitive) often foll by off. to diverge from the main way, road, topic, etc
n.
v.

branch off
branch out