Is-there a name for the (finite) partially ordered sets (posets) which have no "commutative square" in their Hasse diagram ? In other words, finite posets in which all the intervals are (possibly empty) chains ?
This does not imply that the Hasse diagram of the poset is a tree. For example the poset $({1,2,3,4},\leq)$ with $1\leq 2$, $1\leq 3$, $4\leq 2$ and $4\leq 3$ has this property.
They seem to be called diamond-free posets (see this preprint e.g.); Google the term and you'll find many counting problems related to it. Also (if we consider the Hasse diagram as an ordered graph) multitree seems to be used.