A compendium of covariances and correlation coefficients of coalescent tree properties

TL;DR

This paper analyzes the correlations among various properties of coalescent genealogical trees, revealing high correlations among some features and providing insights into tree shape characteristics under the coalescent model.

Contribution

It provides a comprehensive study of the correlations among 15 pairs of genealogical tree features, highlighting their relationships and limiting behaviors.

Findings

High correlations (>0.849) among tree height, length, internal, and basal branch sums.

Zero limiting correlation between external branch sum and height despite fixed expectations.

Results enhance understanding of coalescent tree shape features.

Abstract

Gene genealogies are frequently studied by measuring properties such as their height ($H$), length ($L$), sum of external branches ($E$), sum of internal branches ($I$), and mean of their two basal branches ($B$), and the coalescence times that contribute to the other genealogical features ($T$). These tree properties and their relationships can provide insight into the effects of population-genetic processes on genealogies and genetic sequences. Here, under the coalescent model, we study the 15 correlations among pairs of features of genealogical trees: $H_n$, $L_n$, $E_n$, $I_n$, $B_n$, and $T_k$ for a sample of size $n$, with $2 \leq k \leq n$. We report high correlations among $H_n$, $L_n$, $I_n,$ and $B_n$, with all pairwise correlations of these quantities having values greater than or equal to $\sqrt{6} [6 \zeta(3) + 6 - \pi^2] / ( \pi \sqrt{18 + 9\pi^2 - \pi^4}) \approx 0.84930$ in the limit as $n \rightarrow \infty$. Although $E_n$ has an expectation of 2 for all $n$ and $H_n$ has expectation 2 in the limit as $n \rightarrow \infty$, their limiting correlation is 0. The results contribute toward understanding features of the shapes of coalescent trees.