a secondary shoot or stem in a woody plant generally smaller than the parent.

Merriam-Webster Online Dictionary
branch (noun)
a natural subdivision of a plant stem , especially a secondary shoot or stem (as a bough) arising from a main axis (as of a tree)
something that extends from or enters into a main body or source as
a) (1) a stream that flows into another usually larger stream - tributary
(2) Southern & Midland - creek
b) a side road or way
c) a slender projection (as the tine of an antler)
d) a distinctive part of a mathematical curve
e) a part of a computer program executed as a result of a program decision
a part of a complex body as
a) a division of a family descending from a particular ancestor
b) an area of knowledge that may be considered apart from related areas - pathology is a branch of medicine
c) (1) a division of an organization
(2) a separate but dependent part of a central organization - the neighborhood branch of the city library
d) a language group less inclusive than a family - the Germanic branch of the Indo-European language family
branch (verb)
intransitive verb
to put forth - branches ramify
to spring out (as from a main stem) - diverge
to be an outgrowth - used with from poetry that branched from religious prose
to extend activities - usually used with out the business is branching out
transitive verb
to follow one of two or more branches (as in a computer program)
to ornament with designs of branches
to divide up - section
branch (Wikipedia)
Tree branches of several sizes.
The branches and leaves of a tree.
Looking up into the branch structure of a Pinus sylvestris tree
Leafless tree branches during winter
Exploitation of evergreen bough at the Golden Steinrueck, Vogelsberg

A branch (UK: /ˈbrɑːn/ or UK: /ˈbræn/, US: /ˈbræn/) or tree branch (sometimes referred to in botany as a ramus) is a woody structural member connected to but not part of the central trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term "twig" often refers to a terminus, while "bough" refers only to branches coming directly from the trunk.

Due to a broad range of species of trees, branches and twigs can be found in many different shapes and sizes. While branches can be nearly horizontal, vertical, or diagonal, the majority of trees have upwardly diagonal branches. A number of mathematical properties are associated with tree branchings: they are natural examples of fractal patterns in nature, and, as observed by Leonardo da Vinci, their cross-sectional areas closely follow the da Vinci branching rule.

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