branch
A subset $B$ of a tree $$ is a branch if $B$ is a maximal linearly ordered^{} subset of $T$. That is:

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$$ is a linear ordering of $B$

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If $t\in T\setminus B$ then $B\cup \{t\}$ 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.
Title  branch 

Canonical name  Branch 
Date of creation  20130322 12:52:22 
Last modified on  20130322 12:52:22 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  4 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 05C05 
Classification  msc 03E05 
Related topic  TreeSetTheoretic 
Related topic  ExampleOfTreeSetTheoretic 
Defines  branch 
Defines  cofinal branch 