Definition:Tree (Set Theory)

Definition
A tree is an ordered set $\left({T, \le}\right)$ such that for every $t \in T$, the lower closure of $t$:


 * $\bar\downarrow t = \left\{{s \in T: s \le t}\right\}$

is well-ordered by $\le$.

Branch
A branch of a tree is a maximal chain in it.

Subtree
A subtree of a tree $\left({T, \le}\right)$ is an ordered subset $\left({S, \le}\right)$ with the property that for every $s \in S$ and every $t \in T$ such that $t \le s$, $t \in S$.