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We prove that the number of crossings in a random labeled tree with n vertices in convex position is asymptotically Gaussian with mean \( n^2/6\) and variance \( n^3/45\) . A similar result is proved for points in general position under mild constraints.

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On the Number of Crossings in a Random Labeled Tree

  • 会议论文

  • 发表日期：

    2025年

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  • Octavio Arizmendi,
  • Pilar Cano,
  • Clemens Huemer

摘要

We prove that the number of crossings in a random labeled tree with n vertices in convex position is asymptotically Gaussian with mean \( n^2/6\) and variance \( n^3/45\) . A similar result is proved for points in general position under mild constraints.