Tree Drawings Revisited

Abstract

We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that (1) every tree of size n (with arbitrarily large degree) has a straight-line drawing with area $$n2^{O(\sqrt{\log \log n\log \log \log n})}$$ , improving the longstanding $$O(n\log n)$$ bound; (2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area $$n\sqrt{\log n}(\log \log n)^{O(1)}$$ , improving the longstanding $$O(n\log n)$$ bound; (3) every binary tree of size n has a straight-line orthogonal drawing with area $$n2^{O(\log ^*n)}$$ , improving the previous $$O(n\log \log n)$$ bound; (4) every binary tree of size n has a straight-line order-preserving drawing with area $$n2^{O(\log ^*n)}$$ , improving the previous $$O(n\log \log n)$$ bound; (5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area $$n2^{O(\sqrt{\log n})}$$ , improving the previous $$O(n^{3/2})$$ bound.