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# 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.

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1. 1.

It is not clear to this author if their analysis assumed a much stronger property, that every subtree of size m has degree at most $$O(m^{1/2-\varepsilon })$$.

2. 2.

Constants c in different proofs may be different.

3. 3.

Alternatively, one can see the solution directly without induction: The contribution of the $$(n'/A)s\log ^3 s$$ and $$(n''/s)\log A$$ terms cleary sums to $$O((n/A)s\log ^3 s + (n/s)\log A)$$. The contribution of the first $$\log A$$ term sums to at most $$(2n/A-1)\log A$$, because the number of nodes in the recursion tree is at most $$2n/A-1$$. This is because we can charge at least A units to each leaf and each out-degree-1 node of the recursion tree in such a way that the total number of charges is at most n (since a leaf has $$n\ge A$$, and a out-degree-1 node has $$m=1$$ and $$n-n_1\ge A$$). This implies that the number of leaves and out-degree-1 nodes is at most n / A. The number of nodes of out-degree at least 2 is at most the number of leaves minus 1.

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## Acknowledgements

I thank the reviewers for their careful reading and comments. Funding was provided by National Science Foundation [Grant No. CCF-1814026].

## Author information

Correspondence to Timothy M. Chan.