  A partial order is universal if it contains every countable partial order as
a suborder. In 2017, Fiala, Hubi\v{c}ka, Long and Ne\v{s}et\v{r}il showed that
every interval in the homomorphism order of graphs is universal, with the only
exception being the trivial gap $[K_1,K_2]$. We consider the homomorphism order
restricted to the class of oriented paths and trees. We show that every
interval between two oriented paths or oriented trees of height at least 4 is
universal. The exceptional intervals coincide for oriented paths and trees and
are contained in the class of oriented paths of height at most 3, which forms a
chain.
