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On the asymptotic growth of the number of tree-child networks
European Journal of Combinatorics ( IF 1 ) Pub Date : 2020-12-10 , DOI: 10.1016/j.ejc.2020.103278
Michael Fuchs , Guan-Ru Yu , Louxin Zhang

In a recent paper, McDiarmid, Semple, and Welsh (2015) showed that the number of tree-child networks with n leaves has the factor n2n in its main asymptotic growth term. In this paper, we improve this by completely identifying the main asymptotic growth term up to a constant. More precisely, we show that the number of tree-child networks with n leaves grows like Θn23ea1(3n)1312e2nn2n,where a1=2.338107410 is the largest root of the Airy function of the first kind. For the proof, we bijectively map the underlying graph-theoretical problem onto a problem on words. For the latter, we can find a recurrence to which a recent powerful asymptotic method of Elvey Price, Fang, and Wallner (2019) can be applied.



中文翻译:

关于树子网络数量的渐近增长

McDiarmid,Semple和Welsh(2015)在最近的一篇论文中指出, ñ 叶有因素 ñ2ñ在其主要的渐近增长期。在本文中,我们通过完全确定一个主要的渐近增长项来改善这一点。更确切地说,我们证明了具有ñ 叶子长得像 Θñ-23Ë一种1个3ñ1个312Ë2ññ2ñ哪里 一种1个=-2338107410是第一类Airy函数的最大根源。为了证明这一点,我们将基础的图论问题双射映射到单词问题上。对于后者,我们可以找到一种递归,可以应用Elvey Price,Fang和Wallner(2019)的一种最新的强大渐近方法。

更新日期:2020-12-10
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