Definition:Tree (Set Theory)

Definition
Let $\struct {T, \preceq}$ be an ordered set.

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

is well-ordered by $\preceq$.

Then $\struct {T, \preceq}$ is a tree.