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

Mareike Fischer, Lina Herbst, Sophie Johanna Kersting, Annemarie Luise Kühn and Kristina Wicke
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Mareike Fischer: University of Greifswald, Institute of Mathematics and Computer Science
Lina Herbst: Department of English and American Studies Friedrich Schiller University Jena
Sophie Johanna Kersting: University of Greifswald, Institute of Mathematics and Computer Science
Annemarie Luise Kühn: University of Greifswald, Institute of Mathematics and Computer Science
Kristina Wicke: New Jersey Institute of Technology, Department of Mathematical Sciences

Chapter Chapter 9 in Tree Balance Indices, 2023, pp 89-105 from Springer

Abstract: 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).

Date: 2023
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DOI: 10.1007/978-3-031-39800-1_9

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