  A crossing-free straight-line drawing of a graph is monotone if there is a
monotone path between any pair of vertices with respect to some direction. We
show how to construct a monotone drawing of a tree with $n$ vertices on an
$O(n^{1.5}) \times O(n^{1.5})$ grid whose angles are close to the best possible
angular resolution. Our drawings are convex, that is, if every edge to a leaf
is substituted by a ray, the (unbounded) faces form convex regions. It is known
that convex drawings are monotone and, in the case of trees, also
crossing-free.
  A monotone drawing is strongly monotone if, for every pair of vertices, the
direction that witnesses the monotonicity comes from the vector that connects
the two vertices. We show that every tree admits a strongly monotone drawing.
For biconnected outerplanar graphs, this is easy to see. On the other hand, we
present a simply-connected graph that does not have a strongly monotone drawing
in any embedding.
