What is Branch?

Legal Definition
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. Due to a broad range of species of trees, branches and twigs can be seen in many different shapes and sizes.

While branches can be nearly horizontal, vertical, or diagonal, the majority of trees have upwardly diagonal branches.

The term "twig" often refers to a terminus, while "bough" refers only to branches coming directly from the trunk.
-- Wikipedia
Legal Definition
This is a metaphorical expression, which designates, in the genealogy of a numerous family, a portion of that family which has sprang from the same root or stock; these latter expressions, like the first, are also metaphorical.

2. The whole of a genealogy is often called the genealogical tree; and sometimes it is made to take the form of a tree, which is in the first place divided into as many branches as there are children, afterwards into as many branches as there are grand-children, then of great grandchildren, &c. If, for example, it be desired to have a genealogical tree of Peter's family, Peter will be made the trunk of the tree; if he has had two children, John and James, their names will be written on the first two branches; which will themelves shoot out as many smaller branches as John and James have children; from these other's proceed, till the whole family is represented on the tree; thus the origin, the application, and the use of the word branch in genealogy will be at once perceived.
-- Bouviers Law Dictionary
Legal Definition
An offshoot, lateral extension or subdivision. A branch of a family stock is a group of persons, related among themselves by descent from a common ancestor, and related to the main stock by the fact that that common ancestor descends from the original founder or progenitor.
See also
-- Black's Law Dictionary
Legal Definition
A line of descent from a common ancestor.
-- Ballentine's Law Dictionary