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^{O(√log lognlog log log n)}, improving the longstanding O(nlog 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)^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(nlog logn) 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^{O(log∗ n)}, improving the previous O(nlog 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^O(√logn), improving the O(n^3/2) previous bound by Frati (2007).