TITLE: Terminology about trees
QUESTION [9 upvotes]: In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered.  I am interested in the class of partial orders $P$ such that for every $p \in P$, the set of $q \leq p$ is just linearly ordered. Does this have a name?

REPLY [3 votes]: Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)