  A semiorder is a partially ordered set $P$ with two certain forbidden induced
subposets. This paper establishes a bijection between $n$-element semiorders of
length $H$ and $(n+1)$-node ordered trees of height $H+1$. This bijection
preserves not only the number of elements, but also much additional structure.
Based on this correspondence, we calculate the generating functions and
explicit formulas for the numbers of labeled and unlabeled $n$-element
semiorders of length $H$. We also prove several concise recurrence relations
and provide combinatorial proofs for special cases of the explicit formulas.
