Optimal monotone drawings 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 luw. (Namely, the order of the orthogonal projections of the vertices of P_uw on luw is the same as the order in which 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 log n) - O(n log n). In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most 12n - 12n. The smaller drawing size is achieved by a procedure that carefully assigns primitive vectors to the paths of the input tree T. We also show that there exists a tree T0 such that any monotone drawing of T0 must use a grid of size at least n/9 - n/9. So the size of our monotone drawings is asymptotically optimal.