Tree Drawings Revisited

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 √ log logn log log , 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 √ log n(log log n), improving the longstanding O(n log n) bound; 3. every binary tree of size n has a straight-line orthogonal drawing with area n2 ∗ , improving the previous O(n log log n) bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996); 4. every binary tree of size n has a straight-line order-preserving drawing with area n2 ∗ , improving the previous O(n log log n) bound by Garg and Rusu (2003); 5. every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2 √ , improving the O(n) previous bound by Frati (2007).