Definition:Tree (Set Theory)/Subtree

Definition
Let $\struct {T, \preceq}$ be a tree.

A subtree of $\struct {T, \preceq}$ is an ordered subset $\struct {S, \preceq}$ with the property that:
 * for every $\forall s \in S: \forall t \in T: t \preceq s \implies t \in S$

That is, such that $\struct {S, \preceq}$ is a lower closure of $\struct {T, \preceq}$.