Theoretical diameters of PLR under alternative definitions

Determine the theoretical diameters of the Path-Label Reconciliation (PLR) dissimilarity for the diameter formulations not analyzed in the paper, including the maximum PLR over (i) all species trees with n leaves and pairs of gene trees with m leaves, (ii) a fixed species tree S with gene trees on m leaves, and (iii) a fixed species tree S with specified per-species gene counts ms.

Background

The paper derives the diameter of PLR when the species tree S is fixed and each species has exactly one gene, providing a formula useful for normalization in empirical studies. The authors outline several other natural diameter formulations—varying the species tree, fixing S with more general gene counts per species, etc.—but do not provide theoretical values for these cases.

Completing these diameter analyses would enable consistent normalization across broader reconciliation settings and improve interpretability of PLR in heterogeneous datasets.