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$$ \widehat{s} $$ -shape statistic

In: Tree Balance Indices

Author

Listed:
  • Mareike Fischer

    (University of Greifswald, Institute of Mathematics and Computer Science)

  • Lina Herbst

    (Department of English and American Studies Friedrich Schiller University Jena
    Max Planck Institute of Geoanthropology (formerly MPI for the Science of Human History) Transmission, Infection, Diversification and Evolution Group)

  • Sophie Johanna Kersting

    (University of Greifswald, Institute of Mathematics and Computer Science)

  • Annemarie Luise Kühn

    (University of Greifswald, Institute of Mathematics and Computer Science
    University Medicine Greifswald, Department of Psychiatry and Psychotherapy)

  • Kristina Wicke

    (New Jersey Institute of Technology, Department of Mathematical Sciences)

Abstract

The basic idea behind the $$ \widehat{s} $$ -shape statistic is on the one hand similar to the one for example behind the Sackin index, cf. Chapter 5: It considers all subtree sizes induced by inner vertices, subtracts 1 and sums up the logarithm of the resulting values (and thus effectively downscales the contribution of the larger subtrees to the sum) – so in some sense (considering the monotonicity of the logarithm), it is merely based on a sum of pending subtree sizes. Just as with the Sackin index, the sum used by the $$ \widehat{s} $$ -shape statistic is smallest when all leaves are as close to the root as possible, which meets the intuitive balance concept of a classic scale, cf. Chapter 1. On the other hand, as we point out in the comments of the following fact sheet, the $$ \widehat{s} $$ -shape statistic also has a close relationship to the Yule model (which probably also explains the subtraction of 1 in each summand).

Suggested Citation

  • Mareike Fischer & Lina Herbst & Sophie Johanna Kersting & Annemarie Luise Kühn & Kristina Wicke, 2023. "$$ \widehat{s} $$ -shape statistic," Springer Books, in: Tree Balance Indices, chapter 0, pages 89-105, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-39800-1_9
    DOI: 10.1007/978-3-031-39800-1_9
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