Embedding a balanced binary tree on a bounded point set

Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P , a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are polygonal chains surrounded by P . A special case of the embedding problem is that in which G is a balanced binary tree. In this paper, we present a new algorithm for embedding an n-vertex balanced binary tree BBT on a set S of n points inside a simple m-gon P in O(m + nlog n + mn) time with at most O(m) bends per edge.