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The B2 index of galled trees
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In recent years, there has been an effort to extend the classical notion ofphylogenetic balance, originally defined in the context of trees, to networks.One of the most natural ways to do this is with the so-called $B_2$index. In this paper, we study the $B_2$ index for a prominent class ofphylogenetic networks: galled trees. We show that the $B_2$ index of auniform leaf-labeled galled tree converges in distribution as the networkbecomes large. We characterize the corresponding limiting distribution, andshow that its expected value is 2.707911858984... This is the first timethat a balance index has been studied to this level of detail for a randomphylogenetic network.One specificity of this work is that we use two different and independentapproaches, each with its advantages: analytic combinatorics, and locallimits. The analytic combinatorics approach is more direct, as it relies onstandard tools; but it involves slightly more complex calculations. Becauseit has not previously been used to study such questions, the local limitapproach requires developing an extensive framework beforehand; however, thisframework is interesting in itself and can be used to tackle other similarproblems.
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