Definition:Tree (Set Theory)

Definition
Let $\left({T, \preceq}\right)$ be an ordered set.

Let $\left({T, \preceq}\right)$ be such that for every $t \in T$, the lower closure of $t$:
 * $t^\preceq := \left\{{s \in T: s \preceq t}\right\}$

is well-ordered by $\preceq$.

Then $\left({T, \preceq}\right)$ is a tree.

Branch
A branch of a tree $T$ is a maximal chain in $T$.

Subtree
A subtree of a tree $\left({T, \preceq}\right)$ is an ordered subset $\left({S, \preceq}\right)$ with the property that:
 * for every $\forall s \in S: \forall t \in T: t \preceq s \implies t \in S$