branch
A subset of a tree is a branch if is a maximal linearly ordered subset of . That is:
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is a linear ordering of
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If then is not linearly ordered by .
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.