A subset $B$ of a tree $(T,<_T)$ is a branch if $B$ is a maximal linearly ordered subset of $T$ That is:

• $<_T$ is a linear ordering of $B$
• If $t\in T\setminus B$ then $B\cup \{t\}$ is not linearly ordered by $<_T$

This is the same as the intuitive conception of a branch: it is a set of nodes starting at the root and going all the way to the tip (in infinite sets the conception is more complicated, since there may not be a tip, but the idea is the same). Since branches are maximal there is no way to add an element to a branch and have it remain a branch.

A cofinal branch is a branch which intersects every level of the tree.