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 $$n2^{O(\sqrt{\log \log n\log \log \log n})}$$ n 2 O ( log log n log log log n ) , improving the longstanding $$O(n\log n)$$ O ( n log n ) bound; (2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area $$n\sqrt{\log n}(\log \log n)^{O(1)}$$ n log n ( log log n ) O ( 1 ) , improving the longstanding $$O(n\log n)$$ O ( n log n ) bound; (3) every binary tree of size n has a straight-line orthogonal drawing with area $$n2^{O(\log ^*n)}$$ n 2 O ( log ∗ n ) , improving the previous $$O(n\log \log n)$$ O ( n log log n ) bound; (4) every binary tree of size n has a straight-line order-preserving drawing with area $$n2^{O(\log ^*n)}$$ n 2 O ( log ∗ n ) , improving the previous $$O(n\log \log n)$$ O ( n log log n ) bound; (5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area $$n2^{O(\sqrt{\log n})}$$ n 2 O ( log n ) , improving the previous $$O(n^{3/2})$$ O ( n 3 / 2 ) bound.

中文翻译：

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重新审视树图

我们在一些关于在网格上绘制树木的面积要求的开放问题上取得了进展。我们证明 (1) 每棵大小为 n（具有任意大度数）的树都有一个面积为 $$n2^{O(\sqrt{\log \log n\log \log \log n})} 的直线图$$ n 2 O ( log log n log log log n ) ，改善长期存在的 $$O(n\log n)$$ O ( n log n ) 界限；(2) 每棵大小为n（任意大度数）的树都有一条直线向上绘制，面积为$$n\sqrt{\log n}(\log \log n)^{O(1)}$$ n log n ( log log n ) O ( 1 ) ，改善长期存在的 $$O(n\log n)$$O ( n log n ) 界限；(3) 每个大小为 n 的二叉树都有一个面积为 $$n2^{O(\log ^*n)}$$ n 2 O ( log ∗ n ) 的直线正交图，改进了之前的 $$O( n\log \log n)$$ O ( n log log n ) 绑定；(4) 每个大小为 n 的二叉树都有一个面积为 $$n2^{O(\log ^*n)}$$ n 2 O ( log ∗ n ) 的直线保序图，改进了之前的 $$ O(n\log \log n)$$ O ( n log log n ) 绑定；(5) 每个大小为 n 的二叉树都有一个面积为 $$n2^{O(\sqrt{\log n})}$$ n 2 O ( log n ) 的直线正交保序图，改进了之前的$$O(n^{3/2})$$ O ( n 3 / 2 ) 界限。