Drawing a rooted tree as a rooted y-monotone minimum spanning tree

Given a set P of points in the plane, and a point r is an element of P identified as the root of P, the rooted y-Monotone Minimum Spanning Tree (rooted y-MMST) of P is the connected spanning geometric graph of P in which all the vertices are connected to the root by some y-monotone path and the sum of the Euclidean lengths of its edges is the minimum. We give a linear-time algorithm that draws any rooted tree as a rooted y-MMST. A corollary of the previous sentence is that for every natural number M there exists a rooted y-MMST of maximum degree M. We also show that there exist n-node rooted trees for which any grid drawing as a rooted y-MMST requires a grid of area exponential in n. (C) 2020 Elsevier B.V. All rights reserved.