  We study upward planar straight-line embeddings (UPSE) of directed trees on
given point sets. The given point set $S$ has size at least the number of
vertices in the tree. For the special case where the tree is a path $P$ we show
that: (a) If $S$ is one-sided convex, the number of UPSEs equals the number of
maximal monotone paths in $P$. (b) If $S$ is in general position and $P$ is
composed by three maximal monotone paths, where the middle path is longer than
the other two, then it always admits an UPSE on $S$. We show that the decision
problem of whether there exists an UPSE of a directed tree with $n$ vertices on
a fixed point set $S$ of $n$ points is NP-complete, by relaxing the
requirements of the previously known result which relied on the presence of
cycles in the graph, but instead fixing position of a single vertex. Finally,
by allowing extra points, we guarantee that each directed caterpillar on $n$
vertices and with $k$ switches in its backbone admits an UPSE on every set of
$n 2^{k-2}$ points.
