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 n2O(loglognlogloglogn), 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 nlogn(loglogn)O(1), improving the longstanding O(nlog n) bound; (3) every binary tree of size n has a straight-line orthogonal drawing with area n2O(log∗n), improving the previous O(nlog log n) bound; (4) every binary tree of size n has a straight-line order-preserving drawing with area n2O(log∗n), improving the previous O(nlog log n) bound; (5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2O(logn), improving the previous O(n^{3 / 2}) bound.