### References & Citations

# Quantitative Biology > Populations and Evolution

# Title: A compendium of covariances and correlation coefficients of coalescent tree properties

(Submitted on 24 Mar 2021 (v1), last revised 18 Nov 2021 (this version, v2))

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.

## Submission history

From: Egor Alimpiev [view email]**[v1]**Wed, 24 Mar 2021 19:52:43 GMT (278kb,D)

**[v2]**Thu, 18 Nov 2021 21:55:48 GMT (116kb,D)

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