  We consider drawings of graphs in the plane in which vertices are assigned
distinct points in the plane and edges are drawn as simple curves connecting
the vertices and such that the edges intersect only at their common endpoints.
There is an intuitive quality measure for drawings of a graph that measures the
height of a drawing $\phi : G \rightarrow \mathbb{R}^2$ as follows. For a
vertical line $\ell$ in $\mathbb{R}^2$, let the height of $\ell$ be the
cardinality of the set $\ell \cap \phi(G)$. The height of a drawing of $G$ is
the maximum height over all vertical lines. In this paper, instead of abstract
graphs, we fix a drawing and consider plane graphs. In other words, we are
looking for a homeomorphism of the plane that minimizes the height of the
resulting drawing. This problem is equivalent to the homotopy height problem in
the plane, and the homotopic Fr\'echet distance problem. These problems were
recently shown to lie in NP, but no polynomial-time algorithm or NP-hardness
proof has been found since their formulation in 2009. We present the first
polynomial-time algorithm for drawing trees with optimal height. This
corresponds to a polynomial-time algorithm for the homotopy height where the
triangulation has only one vertex (that is, a set of loops incident to a single
vertex), so that its dual is a tree.
