Tree Drawings Revisited. (arXiv:1803.07185v1 [cs.CG])

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 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 $O(n\log\log n)$ bound by Garg and Rusu (2003);

5. every binary tree of size $n$ has a straight-line orthogonal order-preserving drawing with area $n2^{O(\sqrt{\log n})}$, improving the $O(n^{3/2})$ previous bound by Frati (2007).

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