Drawing a Tree as a Minimum Spanning Tree Approximation

We introduce and study (1 + epsilon)-EMST drawings, i.e. planar straightline drawings of trees such that, for any fixed epsilon > 0, the distance between any two vertices is at least the length of the longest edge in the path connecting them. (1 + e)-EMST drawings are good approximations of Euclidean minimum spanning trees. While it is known that only trees with bounded degree have a Euclidean minimum spanning tree realization, we show that every tree T has a (1 + epsilon)-EMST drawing for any given epsilon > 0. We also present drawing algorithms that compute (1 + epsilon)-EMST drawings of trees with bounded degree in polynomial area. As a byproduct of one of our techniques, we improve the best known area upper bound for Euclidean minimum spanning tree realizations of complete binary trees.