  Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$
points in the plane with no two points on the same horizontal or vertical line.
It is an open question whether $T$ always has a planar drawing on $P$ such that
each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By
giving new methods for drawing trees, we improve the bounds on the size of the
point set $P$ for which such drawings are possible to: $O(n^{1.55})$ for
maximum degree 4 trees; $O(n^{1.22})$ for maximum degree 3 (binary) trees; and
$O(n^{1.142})$ for perfect binary trees.
  Drawing ordered trees with L-shaped edges is harder---we give an example that
cannot be done and a bound of $O(n \log n)$ points for L-shaped drawings of
ordered caterpillars, which contrasts with the known linear bound for unordered
caterpillars.
