  We show that if $D$ is an $n$-vertex digraph with more than $(k-1)n$ arcs
that does not contain any of three forbidden digraphs, then $D$ contains every
antidirected tree on $k$ arcs. The forbidden digraphs are those orientations of
$K_{2, \lceil k/12\rceil}$ where each of the vertices in the class of size two
has either out-degree $0$ or in-degree $0$. This proves a conjecture of
Addario-Berry et al. for a broad class of digraphs, and generalises a result
for $K_{2, \lfloor k/12\rfloor}$-free graphs by Balasubramanian and Dobson.
  We also show that every digraph $D$ on $n$ vertices with more than $(k-1)n$
arcs contains every antidirected $k$-arc caterpillar, thus solving the above
conjecture for caterpillars. This generalises a result of Perles.
