  An L-shaped embedding of a tree in a point set is a planar drawing of the
tree where the vertices are mapped to distinct points and every edge is drawn
as a sequence of two axis-aligned line segments. There has been considerable
work on establishing upper bounds on the minimum cardinality of a point set to
guarantee that any tree of the same size with maximum degree 4 admits an
L-shaped embedding on the point set. However, no non-trivial lower bound is
known to this date, i.e., no known $n$-vertex tree requires more than $n$
points to be embedded. In this paper, we present the first examples of
$n$-vertex trees for $n\in\{13,14,16,17,18,19,20\}$ that require strictly more
points than vertices to admit an L-shaped embedding. Moreover, using computer
help, we show that every tree on $n\leq 12$ vertices admits an L-shaped
embedding in every set of $n$ points. We also consider embedding ordered trees,
where the cyclic order of the neighbors of each vertex in the embedding is
prescribed. For this setting, we determine the smallest non-embeddable ordered
tree on $n=10$ vertices, and we show that every ordered tree on $n\leq 9$ or
$n=11$ vertices admits an L-shaped embedding in every set of $n$ points. We
also construct an infinite family of ordered trees which do not always admit an
L-shaped embedding, answering a question raised by Biedl, Chan, Derka, Jain,
and Lubiw.
