Tree (set theory)

A branch (highlighted green) of a set-theoretic tree. Here dots represent elements, arrows represent the order relation, and ellipses and dashed arrows represent (possibly infinite) un-pictured elements and relationships.

In set theory, a tree is a partially ordered set (T, <) such that for each tT, the set {sT : s < t} is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.