Nearly optimal monotone drawing of trees

A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u,w in G, there exists a path P_uw in G that is monotone in some direction l. (Namely, the order of the orthogonal projections of the vertices of P_uw on l is the same as the order they appear in P_uw.) The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n^1.205)×O(n^1.205). In this paper, we present an algorithm for constructing monotone drawing of trees on a grid of size at most O(nlog⁡n)×O(nlog⁡n).