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

Egor Alimpiev and Noah A Rosenberg∗
Department of Biology, Stanford University, Stanford, CA 94305.

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 $n\rightarrow\infty$, where $\zeta$ is the Riemann zeta
function. Although $E_{n}$ has expectation 2 for all $n$ and $H_{n}$ has
expectation 2 in the $n\rightarrow\infty$ limit, their limiting correlation is
0. The results contribute toward understanding features of the shapes of
coalescent trees.

##  1 Introduction

In coalescent theory, features of gene genealogies are investigated in
relation to the evolutionary processes that are included in population-genetic
models (Hein et al., , 2005; Wakeley, , 2009). For example, comparing a
constant-sized and an exponentially growing population, exponential growth
increases the total length of the branches of a gene genealogy in relation to
its height (Slatkin & Hudson, , 1991; Slatkin, , 1996; Sano & Tachida, ,
2005). Coalescences are rare in recent generations, when the population is
large, and they occur primarily in the period deep in the past when the
population was small.

Several tree features have been used for measuring effects of population-
genetic processes on gene genealogies (Slatkin, , 1996; Uyenoyama, , 1997;
Schierup & Hein, , 2000; Rosenberg, , 2006). For a binary ultrametric tree of
$n$ lineages, these features (Figure 1) include the tree height from the tips
to the root ($H_{n}$), the total length of all the branches ($L_{n}$), the
total length of external branches connecting tips to the nearest internal node
($E_{n}$), the total length of internal branches connecting internal nodes to
other internal nodes ($I_{n}$), and the mean length of the two basal branches
incident to the root node ($B_{n}$).

These tree features can all be expressed as linear combinations, random linear
combinations in some cases, of the same underlying random variables — the
coalescence times $T_{k}$ for coalescence of $k$ to $k-1$ lineages, with
$2\leq k\leq n$. Hence, the quantities are correlated. For example, the tree
height $H_{n}$ includes the mean basal branch length $B_{n}$, and the total
length $L_{n}$ is the sum of the length $E_{n}$ of the external branches and
the length $I_{n}$ of the internal branches; an increase in $L_{n}$
necessarily increases $E_{n}$, $I_{n}$, or both.

Analyses of coalescent models have examined some of the correlations between
tree features, notably the relationship between $H_{n}$ and $L_{n}$ (Fu, ,
1996; Griffiths & Tavaré, , 1996; Rosenberg & Hirsh, , 2003; Arbisser et al.,
, 2018). Here, we perform a detailed investigation of correlations among
$H_{n}$, $L_{n}$, $E_{n}$, $I_{n}$, and $B_{n}$. For each pair, under the
coalescent, assuming a constant-sized population, we evaluate their covariance
and correlation. We explore limiting values as $n\rightarrow\infty$. The
approach follows Arbisser et al.,  (2018), who obtained the covariance and
correlation of $H_{n}$ and $L_{n}$; we perform analogous calculations for all
10 pairs among $\\{H_{n},L_{n},E_{n},I_{n},B_{n}\\}$, as well as for the five
pairs involving one of $\\{H_{n},L_{n},E_{n},I_{n},B_{n}\\}$ and $T_{k}$.

##  2 Tree properties

We consider the standard coalescent model of a constant-sized population of
size $N$ haploids. Time is measured in units of the population size, with one
time unit representing $N$ generations. For sample size $n\geq 2$, we examine
tree properties $H_{n}$, $L_{n}$, $E_{n}$, $I_{n}$, and $B_{n}$, as well as
the coalescence time $T_{k}$, $2\leq k\leq n$. In this section, we recall
basic features of the various quantities.

For convenience, for a mathematical expression we will use frequently, we
write

$S_{p,n}=\sum_{k=1}^{n}\frac{1}{k^{p}}.$ (1)

The limit $S_{p,\infty}=\lim_{n\to\infty}S_{p,n}$ is the Riemann zeta function
$\zeta(p)$. The harmonic sum $S_{1,\infty}$ diverges, and the sum of the
reciprocals of squares is $S_{2,\infty}=\pi^{2}/6\approx 1.64493$. The sum of
the reciprocals of cubes is Apéry’s constant, $S_{3,\infty}=\zeta(3)\approx
1.20206$.

###  2.1 $T_{k}$

$T_{k}$ is a random variable representing the time during which $k$ lineages
coalesce to $k-1$ lineages. The $T_{k}$, $2\leq k\leq n$, are independent and
exponentially distributed with probability density function
$f_{T_{k}}(t_{k})=\binom{k}{2}e^{-\binom{k}{2}t_{k}}$ (Wakeley, , 2009, p.
60). The expectation and variance of $T_{k}$ are then

$\displaystyle\mathbb{E}\left[T_{k}\right]$ $\displaystyle=$
$\displaystyle\frac{2}{k(k-1)},$ (2)
$\displaystyle\operatorname{Var}\left[T_{k}\right]$ $\displaystyle=$
$\displaystyle\frac{4}{k^{2}(k-1)^{2}}.$ (3)

As $n,k\rightarrow\infty$ with $k\leq n$, both $\mathbb{E}\left[T_{k}\right]$
and $\operatorname{Var}\left[T_{k}\right]$ have limit 0.

###  2.2 $H_{n}$

For $n\geq 2$, the height $H_{n}$ of a tree from root to leaves can be written

$H_{n}=\sum_{k=2}^{n}T_{k}.$ (4)

The expectation and variance of $H_{n}$ are then found using the expectation
and variance of $T_{k}$ (eqs. 2 and 3), noting that the $T_{k}$ are
independent:

$\displaystyle\mathbb{E}\left[H_{n}\right]$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{n}\mathbb{E}\left[T_{k}\right]=\frac{2(n-1)}{n},$
(5) $\displaystyle\operatorname{Var}\left[H_{n}\right]$ $\displaystyle=$
$\displaystyle
8\left(\sum_{k=2}^{n}\frac{1}{k^{2}}\right)-4\left(\frac{n-1}{n}\right)^{2}.$
(6)

The variance can be written
$\operatorname{Var}\left[H_{n}\right]=4(2S_{2,n}n^{2}-3n^{2}+2n-1)/n^{2}$. The
limits are $\lim_{n\rightarrow\infty}\mathbb{E}\left[H_{n}\right]=2$ and
$\lim_{n\rightarrow\infty}\operatorname{Var}\left[H_{n}\right]=4\pi^{2}/3-12\approx
1.15947$ (Wakeley, , 2009, p. 76).

###  2.3 $L_{n}$

For $n\geq 2$, the total tree length, summing the lengths of all branches of a
tree, is

$L_{n}=\sum_{k=2}^{n}kT_{k}.$ (7)

By eqs. 2 and 3 and the independence of the $T_{k}$, we have

$\displaystyle\mathbb{E}\left[L_{n}\right]$ $\displaystyle=$
$\displaystyle\sum_{k=2}^{n}k\mathbb{E}\left[T_{k}\right]=2\sum_{k=1}^{n-1}\frac{1}{k},$
(8) $\displaystyle\operatorname{Var}\left[L_{n}\right]$ $\displaystyle=$
$\displaystyle 4\sum_{k=1}^{n-1}\frac{1}{k^{2}}.$ (9)

In terms of $S_{p,n}$ (eq. 1), these expressions are
$\mathbb{E}\left[L_{n}\right]=2S_{1,n-1}$ and
$\operatorname{Var}\left[L_{n}\right]=4S_{2,n-1}$. The limits are
$\lim_{n\rightarrow\infty}\mathbb{E}\left[L_{n}\right]=\infty$ and
$\lim_{n\rightarrow\infty}\operatorname{Var}\left[L_{n}\right]=2\pi^{2}/3\approx
6.57974$ (Wakeley, , 2009, p. 76).

###  2.4 $E_{n}$

The external branches of a tree are the branches that connect leaves to their
nearest internal nodes. Denoting the individual external branch lengths
$e_{1}^{(n)},e_{2}^{(n)},\ldots,e_{n}^{(n)}$, the sum of external branch
lengths is

$E_{n}=e_{1}^{(n)}+e_{2}^{(n)}+\dots+e_{n}^{(n)}.$

The $e_{k}^{(n)}$ are identically distributed, and we write $e_{n}$ for the
length of a randomly chosen external branch of a tree of $n$ lineages. The sum
of the external branches has expectation

$\mathbb{E}\left[E_{n}\right]=n\mathbb{E}\left[e_{n}\right].$ (10)

The random variable $e_{n}$ can be written recursively as (Fu & Li, , 1993,
eq. 7)

$e_{n}=\begin{cases}e_{n-1}+T_{n},&\text{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}with}
probability }\frac{n-2}{n},\\\
T_{n},&\text{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}with}
probability }\frac{2}{n}.\\\ \end{cases}$ (11)

Expressions for $\mathbb{E}\left[e_{n}\right]$,
$\mathbb{E}\left[E_{n}\right]$, and $\operatorname{Var}\left[E_{n}\right]$ can
then be obtained by solving recurrence equations (Fu & Li, , 1993). We have

$\mathbb{E}\left[e_{n}\right]=\frac{2}{n}.$ (12)

For the mean and variance of $E_{n}$, we obtain (Fu & Li, , 1993, eqs. 10 and
14)

$\displaystyle\mathbb{E}\left[E_{n}\right]$ $\displaystyle=$ $\displaystyle
2,$ (13) $\displaystyle\operatorname{Var}\left[E_{n}\right]$ $\displaystyle=$
$\displaystyle\begin{cases}4,&n=2,\\\
\frac{8}{(n-1)(n-2)}[n(\sum_{k=1}^{n-1}\frac{1}{k})-2(n-1)],&n>2.\\\
\end{cases}$ (14)

$\mathbb{E}\left[E_{n}\right]$ is equal to 2 irrespective of the choice of
$n$, so that $\lim_{n\rightarrow\infty}\mathbb{E}\left[E_{n}\right]=2$. The
limit of the variance is
$\lim_{n\rightarrow\infty}\operatorname{Var}\left[E_{n}\right]=0$ (Fu & Li, ,
1993).

###  2.5 $I_{n}$

The internal branches connect internal nodes to other internal nodes. Their
total length is $I_{n}$, with

$I_{n}=L_{n}-E_{n}.$ (15)

The mean and variance of the sum of internal branches are (Fu & Li, , 1993,
eqs. 12 and 17)

$\displaystyle\mathbb{E}\left[I_{n}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[L_{n}\right]-\mathbb{E}\left[E_{n}\right]=2\left(\sum_{k=1}^{n-1}\frac{1}{k}\right)-2,$
(16) $\displaystyle\operatorname{Var}\left[I_{n}\right]$ $\displaystyle=$
$\displaystyle
4\left[\frac{2[S_{1,n-1}n-2(n-1)]}{(n-1)(n-2)}-\frac{2S_{1,n-1}}{n-1}+S_{2,n-1}\right].$
(17)

The limits are $\lim_{n\rightarrow\infty}\mathbb{E}\left[I_{n}\right]=\infty$
and
$\lim_{n\rightarrow\infty}\operatorname{Var}\left[I_{n}\right]=2\pi^{2}/3\approx
6.57974$, the same as for $L_{n}$ (Section 2.3).

###  2.6 $B_{n}$

Finally, we consider the basal branches, the two branches that extend from the
root. We define $B_{n}$ as the mean of the two branch lengths. One of the
branches has length $T_{2}$, and we denote the other length $b_{n}$. We assume
here that $n\geq 4$ for calculations involving $B_{n}$. The appendix of
Uyenoyama,  (1997) gives

$B_{n}=\frac{T_{2}+b_{n}}{2},$ (18)

with

$b_{n}=\bigg{[}\sum_{j=3}^{n-1}\sum_{k=2}^{j}\frac{T_{k}}{j}\prod_{i=3}^{j-1}\left(1-\frac{1}{i}\right)\bigg{]}+\bigg{[}\sum_{k=2}^{n}T_{k}\prod_{i=3}^{n-1}\left(1-\frac{1}{i}\right)\bigg{]}$

for $n\geq 4$. A convenient form for $b_{n}$ encodes the fact that with
probability $2/[j(j-1)]$, $b_{n}=H_{j}$ for $j=3,4,\ldots,n-1$, and with
probability $2/(n-1)$, $b_{n}=H_{n}$:

$b_{n}=\bigg{[}\sum_{j=3}^{n-1}\sum_{k=2}^{j}\frac{2}{j(j-1)}T_{k}\bigg{]}+\bigg{(}\sum_{k=2}^{n}\frac{2}{n-1}T_{k}\bigg{)}.$
(19)

Assuming $n\geq 4$, the branch length $b_{n}$ has expectation (Uyenoyama, ,
1997):

$\mathbb{E}\left[b_{n}\right]=\frac{4}{n}+4\sum_{k=3}^{n-1}\frac{1}{k^{2}}.$
(20)

The expectation and variance of $B_{n}$ then equal

$\displaystyle\mathbb{E}\left[B_{n}\right]$ $\displaystyle=$
$\displaystyle\frac{2}{n}+2\sum_{k=2}^{n-1}\frac{1}{k^{2}},$ (21)
$\displaystyle\operatorname{Var}\left[B_{n}\right]$ $\displaystyle=$
$\displaystyle\frac{2(3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4)}{n^{2}}.$
(22)

The expectation appears in the appendix of Uyenoyama,  (1997). We calculate
the expression for the variance in Section 3.11. Taking limits of these
equations, we obtain
$\lim_{n\rightarrow\infty}\mathbb{E}\left[B_{n}\right]=\pi^{2}/3-2\approx
1.28987$ and
$\lim{n\rightarrow\infty}\operatorname{Var}\left[B_{n}\right]=2+\pi^{2}-\pi^{4}/9\approx
1.04637$.

##  3 Theoretical results

For pairs of variables among $\\{H_{n},L_{n},E_{n},I_{n},B_{n},T_{k}\\}$, we
apply results from Section 2 to compute covariances and correlations. First,
for each pair, we compute their covariance. The covariance together with the
variances of the two quantities from Section 2 provides their correlation. We
obtain the limiting correlation for large trees by taking
$n\rightarrow\infty$. Among the 15 pairs, our analyses for 13 are exact; for
$(E_{n},B_{n})$ and $(I_{n},B_{n})$, we offer approximate covariances and
correlations. We also provide the derivation of eq. 22 for
$\operatorname{Var}\left[B_{n}\right]$.

Note that correlations in pairs involving $E_{n}$ have distinct forms for
$n=2$ and $n\geq 3$, owing to the piecewise definition of
$\operatorname{Var}\left[E_{n}\right]$ in eq. 14. We exclude the case of $n=2$
for pairs involving $I_{n}$, as $I_{2}=0$ with
$\operatorname{Var}\left[I_{2}\right]=0$. We also assume that $B_{n}$ is
defined only for $n\geq 4$.

We present a summary of our mathematical results in Tables 1 and 2. Table 1
shows covariances of pairs of variables and their limits as
$n\rightarrow\infty$. Table 2 shows correlations and their
$n\rightarrow\infty$ limits.

###  3.1 $H_{n}$ and $T_{k}$

We calculate the covariance of $H_{n}$ and $T_{k}$ using
$\operatorname{Cov}[H_{n},T_{k}]=\mathbb{E}\left[H_{n}T_{k}\right]-\mathbb{E}\left[H_{n}\right]\mathbb{E}\left[T_{k}\right]$.
Recalling that $T_{i}$ and $T_{j}$ are independent for $i\neq j$ (Section
2.1), we have
$\mathbb{E}\left[T_{i}T_{j}\right]=\mathbb{E}\left[T_{i}\right]\mathbb{E}\left[T_{j}\right]$
for $i\neq j$. Hence, inserting eq. 4 for $H_{n}$ and eq. 2 for
$\mathbb{E}\left[T_{i}\right]$, we have

$\displaystyle\operatorname{Cov}[H_{n},T_{k}]$
$\displaystyle=\mathbb{E}\left[T_{k}\sum_{i=2}^{n}T_{i}\right]-\mathbb{E}\left[\sum_{i=2}^{n}T_{i}\right]\mathbb{E}\left[T_{k}\right]$
$\displaystyle=\sum_{i=2}^{n}\mathbb{E}\left[T_{i}T_{k}\right]-\sum_{i=2}^{n}\mathbb{E}\left[T_{i}\right]\mathbb{E}\left[T_{k}\right]$
$\displaystyle=\operatorname{Var}\left[T_{k}\right]+\sum_{i=2,\,i\neq
k}^{n}\mathbb{E}\left[T_{i}\right]\mathbb{E}\left[T_{k}\right]-\sum_{i=2,\,i\neq
k}^{n}\mathbb{E}\left[T_{i}\right]\mathbb{E}\left[T_{k}\right]=\operatorname{Var}\left[T_{k}\right]=\frac{4}{k^{2}(k-1)^{2}},$
(23)

where the last step uses $\operatorname{Var}\left[T_{k}\right]$ from eq. 3. We
observe that the covariance is independent of $n$.

For the correlation coefficient
$\operatorname{Corr}[H_{n},T_{k}]=\operatorname{Cov}[H_{n},T_{k}]/\sqrt{\operatorname{Var}\left[H_{n}\right]\operatorname{Var}\left[T_{k}\right]}$,
applying eq. 6 for $\operatorname{Var}\left[H_{n}\right]$, eq. 3 for
$\operatorname{Var}\left[T_{k}\right]$, and eq. 23 for
$\operatorname{Cov}[H_{n},T_{k}]$, we have

$\displaystyle\operatorname{Corr}[H_{n},T_{k}]$
$\displaystyle=\frac{n}{\sqrt{2S_{2,n}n^{2}-3n^{2}+2n-1}}\frac{1}{k(k-1)}.$
(24)

Taking a limit as $n\to\infty$, we obtain

$\lim_{n\to\infty}\operatorname{Corr}[H_{n},T_{k}]=\frac{\sqrt{3}}{\sqrt{\pi^{2}-9}}\frac{1}{k(k-1)}.$
(25)

The limiting correlation decreases to 0 with $k$ from an initial value of
$\frac{1}{2}[\sqrt{3/(\pi^{2}-9)}]\approx 0.92869$ at $k=2$.

###  3.2 $L_{n}$ and $T_{k}$

For $L_{n}$ and $T_{k}$, applying eqs. 7, 8, and 2, we have for the covariance

$\displaystyle\operatorname{Cov}[L_{n},T_{k}]$
$\displaystyle=\mathbb{E}\left[L_{n}T_{k}\right]-\mathbb{E}\left[L_{n}\right]\mathbb{E}\left[T_{k}\right]$
$\displaystyle=\mathbb{E}\left[T_{k}\sum_{i=2}^{n}iT_{i}\right]-\mathbb{E}\left[\sum_{i=2}^{n}iT_{i}\right]\mathbb{E}\left[T_{k}\right]$
$\displaystyle=\mathbb{E}\left[kT_{k}^{2}\right]-k\mathbb{E}\left[T_{k}\right]^{2}=k\operatorname{Var}\left[T_{k}\right]=\frac{4}{k(k-1)^{2}},$
(26)

where the last step uses eq. 3. The covariance of $L_{n}$ and $T_{k}$, like
$\operatorname{Cov}[H_{n},T_{k}]$ (eq. 23), is independent of $n$.

Now we calculate the correlation coefficient from eqs. 26, 9, and 3:

$\displaystyle\operatorname{Corr}[L_{n},T_{k}]$
$\displaystyle=\frac{1}{\sqrt{S_{2,n-1}}}\frac{1}{k-1}.$ (27)

If we let $n\to\infty$, then this quantity becomes

$\lim_{n\to\infty}\operatorname{Corr}[L_{n},T_{k}]=\frac{\sqrt{6}}{\pi}\frac{1}{k-1}.$
(28)

The limiting correlation decreases to 0 with $k$, starting for $k=2$ at
$\sqrt{6}/\pi\approx 0.77970$.

###  3.3 $H_{n}$ and $L_{n}$

Arbisser et al.,  (2018) reported the covariance and correlation of $H_{n}$
and $L_{n}$. By eq. 4 and the linearity of the covariance,

$\operatorname{Cov}[H_{n},L_{n}]=\sum_{k=2}^{n}\operatorname{Cov}[L_{n},T_{k}].\\\
$

Applying eq. 26, we obtain

$\operatorname{Cov}[H_{n},L_{n}]=4S_{2,n-1}-4+\frac{4}{n}.$ (29)

The limit of the covariance is

$\lim_{n\rightarrow\infty}\operatorname{Cov}[H_{n},L_{n}]=\frac{2\pi^{2}}{3}-4\approx
2.57974.$ (30)

Dividing the covariance in eq. 29 by the square root of the product of eqs. 6
and 9, we obtain

$\operatorname{Corr}[H_{n},L_{n}]=\frac{S_{2,n-1}n-n+1}{\sqrt{S_{2,n-1}(2S_{2,n}n^{2}-3n^{2}+2n-1)}}.$
(31)

The limit is

$\lim_{n\rightarrow\infty}\operatorname{Corr}[H_{n},L_{n}]=\frac{\pi^{2}-6}{\pi\sqrt{2(\pi^{2}-9)}}\approx
0.93399.$ (32)

###  3.4 $H_{n}$ and $E_{n}$

For the covariance
$\operatorname{Cov}[H_{n},E_{n}]=\mathbb{E}\left[H_{n}E_{n}\right]-\mathbb{E}\left[H_{n}\right]\mathbb{E}\left[E_{n}\right]$,
we first note that by eqs. 5 and 13, the second term is simply
$4\left(1-\frac{1}{n}\right)$. Expanding $\mathbb{E}\left[H_{n}E_{n}\right]$
by using the definition of $H_{n}$ (eq. 4) gives us

$\mathbb{E}\left[H_{n}E_{n}\right]=\sum_{i=2}^{n}\mathbb{E}\left[E_{n}T_{i}\right]=n\sum_{i=2}^{n}\mathbb{E}\left[e_{n}T_{i}\right],$

as all external branch lengths are identically distributed (eq. 10).

For integers $k$, $i$ with $2\leq k,i,\leq n$, the external branch length
$e_{k}$, representing the length of a randomly chosen external branch for a
tree with $k$ leaves, and the coalescence time $T_{i}$, satisfy (eq. 11)

$e_{k}T_{i}=\begin{cases}e_{k-1}T_{i}+T_{k}T_{i},&\text{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}with}
probability }\frac{k-2}{k},\\\
T_{k}T_{i},&\text{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}with}
probability }\frac{2}{k},\\\ \end{cases}$ (33)

where for convenience, we write $e_{1}=0$.

Note that $e_{k}$ and $T_{i}$ are independent for $i>k$, as the recurrence for
$e_{k}$ is constructed only using coalescence times $T_{2},T_{3},\ldots,T_{k}$
(eq. 11); each of these times is indepdendent of $T_{i}$ for $i>k$ (Section
2.1). We solve to find $\mathbb{E}\left[e_{n}T_{i}\right]$ by computing
$\mathbb{E}\left[e_{k}T_{i}\right]$, incrementing $k$ from 2 to $n$. The
calculations are similar to those of the Appendix of Fu & Li,  (1993).

$\mathbb{E}\left[e_{2}T_{2}\right]$ is trivial, with $e_{2}=T_{2}$, and
$\mathbb{E}\left[e_{2}T_{2}\right]=\mathbb{E}\left[T_{2}^{2}\right]=2$ by eqs.
2 and 3. By eqs. 12 and 2 and the independence of $e_{k}$ and $T_{i}$ for
$i>k$, for $i\geq 3$,

$\mathbb{E}\left[e_{i-1}T_{i}\right]=\mathbb{E}\left[e_{i-1}\right]\mathbb{E}\left[T_{i}\right]=\frac{4}{i(i-1)^{2}}.$

Noting
$\mathbb{E}\left[T_{i}^{2}\right]=\operatorname{Var}\left[T_{i}\right]+\mathbb{E}\left[T_{i}\right]^{2}=2\mathbb{E}\left[T_{i}\right]^{2}$
by eqs. 2 and 3, we use eq. 33 to write an expression for
$\mathbb{E}\left[e_{i}T_{i}\right]$:

$\mathbb{E}\left[e_{i}T_{i}\right]=\frac{i-2}{i}\mathbb{E}\left[e_{i-1}T_{i}\right]+\mathbb{E}\left[T_{i}^{2}\right]=\frac{4}{i(i-1)^{2}}.$
(34)

Next, incrementing eq. 34, we have

$\mathbb{E}\left[e_{i+1}T_{i}\right]=\frac{i-1}{i+1}\mathbb{E}\left[e_{i}T_{i}\right]+\mathbb{E}\left[T_{i+1}\right]\mathbb{E}\left[T_{i}\right]=\frac{4}{i^{2}(i-1)},$
(35)

by eqs. 2 and 12.

The final step is to solve the recurrence equation

$\mathbb{E}\left[e_{n}T_{i}\right]=\frac{n-2}{n}\mathbb{E}\left[e_{n-1}T_{i}\right]+\mathbb{E}\left[T_{n}\right]\mathbb{E}\left[T_{i}\right],$

with initial condition eq. 35. Recalling the case of $i=n=2$, with $2\leq
i\leq n$, we obtain solution

$\mathbb{E}\left[e_{n}T_{i}\right]=\frac{4}{i(i-1)(n-1)}.$ (36)

Applying eq. 10, the expression for $\operatorname{Cov}[H_{n},E_{n}]$ becomes

$\operatorname{Cov}[H_{n},E_{n}]=n\sum_{i=2}^{n}\frac{4}{i(i-1)(n-1)}-4\left(1-\frac{1}{n}\right)=\frac{4}{n}.$
(37)

The limit of the covariance as $n\rightarrow\infty$ is

$\lim_{n\to\infty}\operatorname{Cov}[H_{n},E_{n}]=0.$ (38)

Dividing eq. 37 by the square root of the product of variances from eqs. 6 and
14, the correlation is

$\displaystyle\operatorname{Corr}[H_{n},E_{n}]=\begin{cases}1,&n=2,\\\
\frac{\sqrt{(n-1)(n-2)}}{\sqrt{2\left(2S_{2,n}n^{2}-3n^{2}+2n-1\right)\left(S_{1,n-1}n-2n+2\right)}},&n>2.\end{cases}$
(39)

The limit of the correlation is

$\lim_{n\to\infty}\operatorname{Corr}[H_{n},E_{n}]=0.$ (40)

###  3.5 $E_{n}$ and $T_{k}$

In the process of computing $\operatorname{Cov}[H_{n},E_{n}]$, we have
obtained an expression for $\mathbb{E}\left[e_{n}T_{i}\right]$ (eq. 36), from
which we can obtain
$\operatorname{Cov}[E_{n},T_{k}]=n\mathbb{E}\left[e_{n}T_{k}\right]-\mathbb{E}\left[E_{n}\right]\mathbb{E}\left[T_{k}\right]$.
Applying eqs. 13 and 2, we have

$\displaystyle\operatorname{Cov}[E_{n},T_{k}]$
$\displaystyle=\frac{4}{k(k-1)(n-1)}.$ (41)

Irrespective of the value of $k$, we have

$\lim_{n\to\infty}\operatorname{Cov}[E_{n},T_{k}]=0.$ (42)

Applying eqs. 3 and 14, the correlation coefficient is

$\displaystyle\operatorname{Corr}[E_{n},T_{k}]$
$\displaystyle=\begin{cases}1,&n=2,\\\
\frac{\sqrt{n-2}}{\sqrt{2(n-1)(S_{1,n-1}n-2n+2)}},&n\geq 3.\end{cases}$ (43)

The correlation coefficient is independent of $k$, and it has limit

$\lim_{n\to\infty}\operatorname{Corr}[E_{n},T_{k}]=0.$ (44)

###  3.6 $L_{n}$ and $E_{n}$

Fu & Li,  (1993) provided the expression for
$\mathbb{E}\left[L_{n}E_{n}\right]$ (see also p. 167 of Durrett,  (2008), with
all values scaled by $2N_{e}$). The main result is the following expression,
obtained by solving recurrence equations:

$\mathbb{E}\left[L_{n}E_{n}\right]=\frac{4S_{1,n-1}n}{n-1}.$

We can use this result to calculate the covariance of $L_{n}$ and $E_{n}$ by
$\operatorname{Cov}[L_{n},E_{n}]=\mathbb{E}\left[L_{n}E_{n}\right]-\mathbb{E}\left[L_{n}\right]\mathbb{E}\left[E_{n}\right]$
with eqs. 8 and 13. The covariance can also be quickly obtained from eqs. 7
and 41,

$\operatorname{Cov}[L_{n},E_{n}]=\sum_{k=2}^{n}k\operatorname{Cov}[E_{n},T_{k}]=\frac{4S_{1,n-1}}{n-1}.$
(45)

The limit is

$\lim_{n\to\infty}\operatorname{Cov}[L_{n},E_{n}]=0.$ (46)

Applying eqs. 45, 9, and 14, the correlation coefficient of $L_{n}$ and
$E_{n}$ is

$\displaystyle\operatorname{Corr}[L_{n},E_{n}]$
$\displaystyle=\begin{cases}1,&n=2,\\\
\frac{S_{1,n-1}\sqrt{n-2}}{\sqrt{2S_{2,n-1}(n-1)(S_{1,n-1}n-2n+2)}},&n\geq
3,\end{cases}$ (47)

with the limit

$\lim_{n\to\infty}\operatorname{Corr}[L_{n},E_{n}]=0.$ (48)

###  3.7 $H_{n}$ and $I_{n}$

For the pair $H_{n}$ and $I_{n}$, we exploit results obtained for other pairs
to quickly obtain the covariance. Remembering that $I_{n}=L_{n}-E_{n}$ (eq.
15), we use eqs. 29 and 37 to obtain for $n\geq 3$

$\displaystyle\operatorname{Cov}[H_{n},I_{n}]$
$\displaystyle=\operatorname{Cov}[H_{n},L_{n}]-\operatorname{Cov}[H_{n},E_{n}]$
$\displaystyle=4S_{2,n-1}-4.$ (49)

For this covariance, we have

$\lim_{n\to\infty}\operatorname{Cov}[H_{n},I_{n}]=\frac{2\pi^{2}}{3}-4\approx
2.57974.$ (50)

From the covariance in eq. 49 and variances in eqs. 6 and 17, we compute the
correlation coefficient:

$\operatorname{Corr}[H_{n},I_{n}]=\frac{\left(S_{2,n-1}-1\right)n\sqrt{(n-1)(n-2)}}{\sqrt{\left(2S_{2,n}n^{2}-3n^{2}+2n-1\right)\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}}.$
(51)

The limit is the same as that of $\operatorname{Corr}[H_{n},L_{n}]$, or (eq.
32)

$\lim_{n\to\infty}\operatorname{Corr}[H_{n},I_{n}]=\frac{\pi^{2}-6}{\pi\sqrt{2\left(\pi^{2}-9\right)}}\approx
0.93399.$ (52)

###  3.8 $I_{n}$ and $T_{k}$

By eqs. 15, 26, and 41, assuming $n\geq 3$, we have

$\displaystyle\operatorname{Cov}[I_{n},T_{k}]$
$\displaystyle=\operatorname{Cov}[L_{n},T_{k}]-\operatorname{Cov}[E_{n},T_{k}]$
$\displaystyle=\frac{4(n-k)}{k(k-1)^{2}(n-1)}.$ (53)

The limit of this expression is a rapidly decreasing function of $k$,

$\lim_{n\to\infty}\operatorname{Cov}[I_{n},T_{k}]=\frac{4}{k(k-1)^{2}}.$ (54)

Using the variances in eqs. 17 and 3, the correlation coefficient is

$\operatorname{Corr}[I_{n},T_{k}]=\frac{(n-k)\sqrt{n-2}}{(k-1)\sqrt{(n-1)\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}},$
(55)

with limit

$\lim_{n\to\infty}\operatorname{Corr}[I_{n},T_{k}]=\frac{\sqrt{6}}{\pi}\frac{1}{k-1}.$
(56)

The limit of $\operatorname{Corr}[I_{n},T_{k}]$ as $n\to\infty$ is equal to
that of $\operatorname{Corr}[L_{n},T_{k}]$ (eq. 28),

###  3.9 $L_{n}$ and $I_{n}$

By eq. 15, we can apply eqs. 9 and 45 to obtain for $n\geq 3$

$\displaystyle\operatorname{Cov}[L_{n},I_{n}]$
$\displaystyle=\operatorname{Var}\left[L_{n}\right]-\operatorname{Cov}[L_{n},E_{n}]$
$\displaystyle=4S_{2,n-1}-\frac{4S_{1,n-1}}{n-1}.$ (57)

The limit as $n\to\infty$ is

$\lim_{n\to\infty}\operatorname{Cov}[L_{n},I_{n}]=\frac{2\pi^{2}}{3}\approx
6.57974.$ (58)

For the correlation coefficient, applying eqs. 57, 9, and 17, we get

$\operatorname{Corr}[L_{n},I_{n}]=\frac{[S_{2,n-1}(n-1)-S_{1,n-1}]\sqrt{n-2}}{(n-1)\sqrt{S_{2,n-1}\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}},$
(59)

with

$\lim_{n\to\infty}\operatorname{Corr}[L_{n},I_{n}]=1.$ (60)

###  3.10 $E_{n}$ and $I_{n}$

For this pair, with $n\geq 3$, the covariance was reported by Fu & Li,
(1993):

$\operatorname{Cov}[E_{n},I_{n}]=\frac{4S_{1,n-1}}{n-1}-\frac{8S_{1,n-1}n}{(n-1)(n-2)}+\frac{16}{n-2}.$
(61)

We can also obtain this result quickly from eqs. 15, 57, and 17, as
$\operatorname{Cov}[E_{n},I_{n}]=\operatorname{Cov}[L_{n}-I_{n},I_{n}]=\operatorname{Cov}[L_{n},I_{n}]-\operatorname{Var}[I_{n}]$.
In the limit, we have

$\lim_{n\to\infty}\operatorname{Cov}[E_{n},I_{n}]=0.$ (62)

For the correlation coefficient, we divide eq. 61 by the product of the square
roots of eqs. 14 and 17:

$\operatorname{Corr}[E_{n},I_{n}]=\frac{4(n-1)-S_{1,n-1}(n+2)}{\sqrt{2\left(S_{1,n-1}n-2n+2\right)\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}},$
(63)

with the limit

$\lim_{n\to\infty}\operatorname{Corr}[E_{n},I_{n}]=0.$ (64)

This result is equal to the limit for $\operatorname{Corr}[L_{n},E_{n}]$ (eq.
48).

###  3.11 $\operatorname{Var}\left[B_{n}\right]$

To obtain correlation coefficients involving $B_{n}$, assuming $n\geq 4$, we
first verify the expression for $\operatorname{Var}\left[B_{n}\right]$ in eq.
22. By definition of $B_{n}$ in eq. 18, we have

$\operatorname{Var}\left[B_{n}\right]=\mathbb{E}\left[\frac{1}{4}(T_{2}+b_{n})^{2}\right]-\mathbb{E}\left[\frac{1}{2}(T_{2}+b_{n})\right]^{2}=\frac{1}{4}\operatorname{Var}\left[b_{n}\right]+\frac{1}{2}\operatorname{Cov}[b_{n},T_{2}]+\frac{1}{4},$
(65)

where we have used $\operatorname{Var}\left[T_{2}\right]=1$ (eq. 3).

To calculate $\operatorname{Var}\left[b_{n}\right]$, we first recall that for
$j=3,4,\ldots,n-1$, with probability $2/[j(j-1)]$, we have $b_{n}=H_{j}$; with
probability $2/(n-1)$ we have $b_{n}=H_{n}$. Hence, applying eq. 19 and
$\mathbb{E}\left[H_{k}^{2}\right]=\operatorname{Var}\left[H_{k}\right]+\mathbb{E}\left[H_{k}\right]^{2}$
with eqs. 5 and 6, we have

$\displaystyle\mathbb{E}\left[b_{n}^{2}\right]$
$\displaystyle=\bigg{[}\sum_{j=3}^{n-1}\frac{2}{j(j-1)}\mathbb{E}\left[H_{j}\right]^{2}\bigg{]}+\frac{2}{n-2}\mathbb{E}\left[H_{n}\right]^{2}$
$\displaystyle=\frac{-16S_{2,n-1}n^{2}+30n^{2}-16n-16}{n^{2}}.$

Using the expression for $\mathbb{E}\left[b_{n}\right]$ from eq. 20, we obtain

$\operatorname{Var}\left[b_{n}\right]=\frac{24S_{2,n-1}n^{2}-16S_{2,n-1}^{2}n^{2}+5n^{2}-32S_{2,n-1}n+24n-32}{n^{2}}.$
(66)

Next, we compute $\operatorname{Cov}[b_{n},T_{k}]$ and insert $k=2$. By eq.
19, applying the independence of the $T_{i}$ (Section 2.1) and inserting eq.
3, we have

$\displaystyle\operatorname{Cov}[b_{n},T_{k}]$
$\displaystyle=\bigg{[}\sum_{j=3}^{n-1}\sum_{i=2}^{j}\frac{2}{j(j-1)}\operatorname{Cov}[T_{i},T_{k}]\bigg{]}+\bigg{(}\sum_{i=2}^{n}\frac{2}{n-1}\operatorname{Cov}[T_{i},T_{k}]\bigg{)}$
$\displaystyle=\bigg{[}\sum_{i=2}^{n-1}\sum_{j=i\text{ if }i\geq 3\atop
j=3\text{ if
}i=2}^{n-1}\frac{2}{j(j-1)}\operatorname{Cov}[T_{i},T_{k}]\bigg{]}+\bigg{(}\frac{2}{n-1}\operatorname{Var}\left[T_{k}\right]\bigg{)}$
$\displaystyle=\begin{cases}\left[\sum_{j=3}^{n-1}\frac{2}{j(j-1)}\operatorname{Var}\left[T_{k}\right]\right]+\left(\frac{2}{n-1}\operatorname{Var}\left[T_{k}\right]\right),&k=2,\\\
\left[\sum_{j=k}^{n-1}\frac{2}{j(j-1)}\operatorname{Var}\left[T_{k}\right]\right]+\left(\frac{2}{n-1}\operatorname{Var}\left[T_{k}\right]\right),&k=3,4,\ldots,n-1,\\\
\frac{2}{n-1}\operatorname{Var}\left[T_{k}\right],&k=n,\end{cases}$
$\displaystyle=\begin{cases}1,&k=2,\\\
\frac{8}{k^{2}(k-1)^{3}},&k=3,4,\ldots,n.\end{cases}$ (67)

Inserting $\operatorname{Var}\left[b_{n}\right]$ from eq. 66 and
$\operatorname{Cov}[b_{n},T_{2}]$ from eq. 67 into eq. 65, we confirm eq. 22.

###  3.12 $B_{n}$ and $T_{k}$

We extract $\operatorname{Cov}[B_{n},T_{k}]$ from Section 3.11, as
$\operatorname{Cov}[B_{n},T_{k}]={\operatorname{Cov}[b_{n},T_{k}]}/{2}+{\operatorname{Cov}[T_{2},T_{k}]}/{2}$
by the definition in eq. 18, and
$\operatorname{Cov}[T_{2},T_{k}]=\delta_{k,2}$, where $\delta$ is the
Kronecker delta (Section 2.1). By eq. 67, recalling $n\geq 4$,

$\operatorname{Cov}[B_{n},T_{k}]=\frac{4}{k^{2}(k-1)^{3}}.$ (68)

The covariance is independent of $n$.

For the correlation coefficient, using eqs. 68, 22, and 3, we have

$\operatorname{Corr}[B_{n},T_{k}]=\frac{\sqrt{2}n}{\sqrt{3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4}}\frac{1}{k(k-1)^{2}},$
(69)

with the limit

$\lim_{n\to\infty}\operatorname{Corr}[B_{n},T_{k}]=\frac{6}{\sqrt{18+9\pi^{2}-\pi^{4}}}\frac{1}{k(k-1)^{2}}.$
(70)

The limit begins at $3/\sqrt{18+9\pi^{2}-\pi^{4}}\approx 0.97759$ for $k=2$
and quickly decreases to 0 as $k$ increases.

###  3.13 $H_{n}$ and $B_{n}$

To obtain $\operatorname{Cov}[H_{n},B_{n}]$ with $n\geq 4$, we begin from eq.
18:

$\operatorname{Cov}[H_{n},B_{n}]=\frac{1}{2}\operatorname{Cov}[H_{n},b_{n}]+\frac{1}{2}\operatorname{Cov}[H_{n},T_{2}].$

The second term was computed in eq. 23, $\operatorname{Cov}[H_{n},T_{2}]=1$.

For the first term, $\operatorname{Cov}[H_{n},b_{n}]$, we decompose $H_{n}$
(eq. 4) and apply eq. 67 to obtain

$\displaystyle\operatorname{Cov}[H_{n},b_{n}]$
$\displaystyle=\sum_{k=2}^{n}\operatorname{Cov}[b_{n},T_{k}]$
$\displaystyle=1+\sum_{k=3}^{n}\frac{8}{k^{2}(k-1)^{3}}.$

We use a partial fraction decomposition to sum the series, obtaining

$\displaystyle\operatorname{Cov}[H_{n},B_{n}]$
$\displaystyle=1+\sum_{k=3}^{n}\frac{4}{k^{2}(k-1)^{3}}$
$\displaystyle=\frac{4[S_{3,n-1}n^{2}-3S_{2,n-1}n^{2}+(n-1)(4n+1)]}{n^{2}}.$
(71)

The asymptotic limit of $\operatorname{Cov}[H_{n},B_{n}]$ is

$\lim_{n\to\infty}\operatorname{Cov}[H_{n},B_{n}]=4\zeta(3)+16-2\pi^{2}\approx
1.06902.$ (72)

The correlation coefficient is then equal to:

$\operatorname{Corr}[H_{n},B_{n}]=\frac{\sqrt{2}[S_{3,n-1}n^{2}-3S_{2,n-1}n^{2}+(n-1)(4n+1)]}{\sqrt{(2S_{2,n}n^{2}-3n^{2}+2n-1)(3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4})}.$
(73)

The limit of the correlation coefficient is:

$\lim_{n\to\infty}\operatorname{Corr}[H_{n},B_{n}]=\frac{3\sqrt{3}[2\zeta(3)+8-\pi^{2}]}{\sqrt{(\pi^{2}-9)(18+9\pi^{2}-\pi^{4})}}\approx
0.97054.$ (74)

###  3.14 $L_{n}$ and $B_{n}$

In a manner similar to that used in Section 3.13, with $n\geq 4$, we expand
$\operatorname{Cov}[L_{n},B_{n}]$ using eq. 18:

$\operatorname{Cov}[L_{n},B_{n}]=\frac{1}{2}\operatorname{Cov}[L_{n},b_{n}]+\frac{1}{2}\operatorname{Cov}[L_{n},T_{2}].$

The second term is $\operatorname{Cov}[L_{n},T_{2}]=2$ by eq. 26. The first
term is decomposable by eq. 7; applying eq. 67,

$\displaystyle\operatorname{Cov}[L_{n},b_{n}]$
$\displaystyle=\sum_{k=2}^{n}k\operatorname{Cov}[b_{n},T_{k}]$
$\displaystyle=2+\sum_{k=3}^{n}\frac{8}{k(k-1)^{3}}.$

Summing the series, we have

$\displaystyle\operatorname{Cov}[L_{n},B_{n}]$
$\displaystyle=2+\sum_{k=3}^{n}\frac{4}{k(k-1)^{3}}$
$\displaystyle=\frac{4[S_{3,n-1}n-S_{2,n-1}n+n-1]}{n}.$ (75)

The limiting covariance is

$\lim_{n\to\infty}\operatorname{Cov}[L_{n},B_{n}]=4\zeta(3)+4-\frac{2\pi^{2}}{3}\approx
2.22849.$ (76)

Using eqs. 75, 9, and 22, we now obtain an expression for the correlation
coefficient:

$\operatorname{Corr}[L_{n},B_{n}]=\frac{\sqrt{2}(S_{3,n-1}n-S_{2,n-1}n+n-1)}{\sqrt{S_{2,n-1}({3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4})}}.$
(77)

The limit is

$\lim_{n\to\infty}\operatorname{Corr}[L_{n},B_{n}]=\frac{\sqrt{6}[6\zeta(3)+6-\pi^{2}]}{\pi\sqrt{18+9\pi^{2}-\pi^{4}}}\approx
0.84930.$ (78)

###  3.15 $E_{n}$ and $B_{n}$

For $\operatorname{Cov}[E_{n},B_{n}]$, we obtain an approximate rather than
exact answer. Decomposing $B_{n}$ by eq. 18, we have

$\operatorname{Cov}[E_{n},B_{n}]=\frac{1}{2}\operatorname{Cov}[E_{n},T_{2}]+\frac{1}{2}\operatorname{Cov}[E_{n},b_{n}].$
(79)

Recall that $b_{n}$ can be defined conditionally, in terms of a random
variable $J$ that characterizes the coalescence times that it contains
(Section 2.6). More precisely, we say that for a random variable $J$,
$b_{n}=H_{j}$ with probability $p_{j}$, where
$p_{j}=\mathbb{P}[J=j]=2/[j(j-1)]$ for $J=3,4,\ldots,n-1$ and $p_{j}=2/(j-1)$
for $J=n$. We can then decompose the covariance
$\operatorname{Cov}[E_{n},b_{n}]$ by the conditional covariance formula,
conditioning on $J$:

$\operatorname{Cov}[E_{n},b_{n}]=\mathbb{E}\bigg{[}\operatorname{Cov}[E_{n},b_{n}|J]\bigg{]}+\operatorname{Cov}\bigg{[}\mathbb{E}\left[E_{n}|J\right],\mathbb{E}\left[B_{n}|J\right]\bigg{]}.$
(80)

We next perform an approximation by ignoring the second term in the covariance
decomposition. Noting that $\operatorname{Cov}[E_{n},T_{2}]=\frac{2}{n-1}$ by
eq. 41, we use eq. 79 together with eq. 80 to write approximations

$\displaystyle\widetilde{\operatorname{Cov}}\left[E_{n},b_{n}\right]$
$\displaystyle=\mathbb{E}\bigg{[}\operatorname{Cov}[E_{n},b_{n}|J]\bigg{]}$
(81) $\displaystyle\widetilde{\operatorname{Cov}}\left[E_{n},B_{n}\right]$
$\displaystyle=\frac{1}{n-1}+\frac{1}{2}\widetilde{\operatorname{Cov}}\left[E_{n},b_{n}\right].$
(82)

Weighting each $\operatorname{Cov}[H_{j},E_{n}]$ by the associated probability
$p_{j}$, and decomposing $H_{j}$ by eq. 4, eq. 81 gives

$\displaystyle\mathbb{E}\bigg{[}\operatorname{Cov}[E_{n},b_{n}|J]\bigg{]}$
$\displaystyle=\sum_{j=3}^{n}p_{j}\operatorname{Cov}[E_{n},b_{n}|J=j]$
$\displaystyle=\bigg{[}\sum_{j=3}^{n-1}\frac{2}{j(j-1)}\operatorname{Cov}[H_{j},E_{n}]\bigg{]}+\frac{2}{n-1}\operatorname{Cov}[H_{n},E_{n}]$
$\displaystyle=\bigg{[}\sum_{j=3}^{n-1}\frac{2}{j(j-1)}\sum_{k=2}^{j}\operatorname{Cov}[E_{n},T_{k}]\bigg{]}+\frac{2}{n-1}\sum_{k=2}^{j}\operatorname{Cov}[E_{n},T_{k}].$

We can then insert the result of eq. 41 and simplify to obtain

$\widetilde{\operatorname{Cov}}\left[E_{n},b_{n}\right]=\frac{2(4S_{2,n-1}n-5n+4)}{n(n-1)}.$
(83)

Finally, inserting eq. 83 into eq. 82, we have

$\widetilde{\operatorname{Cov}}\left[E_{n},B_{n}\right]=\frac{4(S_{2,n-1}n-n+1)}{n(n-1)}.$
(84)

The limit is

$\lim_{n\rightarrow\infty}\widetilde{\operatorname{Cov}}\left[E_{n},B_{n}\right]=0.$
(85)

For the approximate correlation coefficient
$\widetilde{\operatorname{Corr}}\left[E_{n},B_{n}\right]=\widetilde{\operatorname{Cov}}\left[E_{n},B_{n}\right]/\sqrt{\operatorname{Var}\left[E_{n}\right]\operatorname{Var}\left[B_{n}\right]}$,
we use eqs. 84, 14, and 22 to obtain

$\widetilde{\operatorname{Corr}}\left[E_{n},B_{n}\right]=\frac{(S_{2,n-1}n-n+1)\sqrt{n-2}}{\sqrt{(n-1)(S_{1,n-1}n-2n+2)(3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4)}},$
(86)

with limit

$\lim_{n\rightarrow\infty}\widetilde{\operatorname{Corr}}\left[E_{n},B_{n}\right]=0.$
(87)

###  3.16 $I_{n}$ and $B_{n}$

We use eq. 15 and results involving $L_{n}$ (eq. 75) and $E_{n}$ (eq. 84) to
obtain

$\displaystyle\widetilde{\operatorname{Cov}}\left[I_{n},B_{n}\right]$
$\displaystyle=\operatorname{Cov}[L_{n},B_{n}]-\widetilde{\operatorname{Cov}}\left[E_{n},B_{n}\right]$
$\displaystyle=\frac{4(S_{3,n-1}n-S_{2,n-1}n+n-S_{3,n-1}-1)}{n-1},$ (88)

with limit

$\lim_{n\to\infty}\widetilde{\operatorname{Cov}}\left[I_{n},B_{n}\right]=4\zeta(3)+4-\frac{2\pi^{2}}{3}\approx
2.22849.$ (89)

Dividing eq. 88 by the product of the square roots of eqs. 17 and 22, the
approximate correlation is

$\widetilde{\operatorname{Corr}}\left[I_{n},B_{n}\right]=\frac{\sqrt{2}(S_{3,n-1}n-S_{2,n-1}n+n-S_{3,n-1}-1)n\sqrt{n-2}}{\sqrt{(n-1)[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)](3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4)}},$
(90)

with limit

$\lim_{n\to\infty}\widetilde{\operatorname{Corr}}\left[I_{n},B_{n}\right]=\frac{\sqrt{6}[6\zeta(3)+6-\pi^{2}]}{\pi\sqrt{18+9\pi^{2}-\pi^{4}}}\approx
0.84930.$ (91)

##  4 Numerical and simulation-based analysis

###  4.1 Analysis methods

We examine the results of Section 3 summarized in Tables 1 and 2 numerically
and by coalescent simulation. For 13 of 15 covariances and correlations, the
theoretical results are exact, and simulations merely verify that the
mathematics has proceeded without error. For the covariances and correlations
involving $(E_{n},B_{n})$ and $(I_{n},B_{n})$, the theoretical results are
approximate, and the simulations assess the accuracy of the approximations.

We simulated the coalescent process for a series of values of $n$ beginning
with $n=2$, at each value of $n$ performing 100,000 replicate simulations. To
generate the simulated replicates, we employed ms (Hudson, , 2002), using the
command ms n 100000 -T, with n taken from $\\{2,3,\ldots,50\\}$. In the set of
simulated replicates, we evaluated simulated covariances and correlation
coeficients for pairs of quantities.

We plot the mathematical results of Tables 1 and 2 together with simulation
values in Figures 2-5. Figures 2 and 3 show covariances of pairs of variables;
Figures 4 and 5 show correlations.

###  4.2 Accuracy of approximations

Figure 2 shows the analytical and simulated covariances, and Figure 4 shows
the analytical and simulated correlations, for pairs of variables among
$\\{H_{n},L_{n},I_{n},E_{n},B_{n}\\}$. For pairs of variables for which no
approximations were needed in obtaining covariances—all except $(E_{n},B_{n})$
and $(I_{n},B_{n})$—the simulated and analytical values produce plots that are
nearly indistinguishable.

For $(E_{n},B_{n})$ and $(I_{n},B_{n})$, the approximate and simulated
correlations are close, but noticeably different (Figure 4); the mean absolute
difference between the analytical and simulated values across choices of $n$
from 4 to 30 is $0.02458$ for $(E_{n},B_{n})$ and $0.01089$ for
$(I_{n},B_{n})$. For covariance, which unlike the correlation coefficient is
not standardized to lie in $[-1,1]$, the approximate and simulated values are
quite close, with corresponding mean absolute deviations of $0.02372$ for
$(E_{n},B_{n})$ and $0.03101$ for $(I_{n},B_{n})$.

###  4.3 Properties of correlations

We observe that $H_{n}$, $L_{n}$, $I_{n}$, and $B_{n}$ all remain strongly
correlated as $n$ increases, with the six limiting correlations among these
four quantities lying between 0.84930 for $\operatorname{Corr}[L_{n},B_{n}]$
and $\operatorname{Corr}[I_{n},B_{n}]$ and 1 for
$\operatorname{Corr}[L_{n},I_{n}]$ (Table 2). The high limiting
$\operatorname{Corr}[H_{n},L_{n}]$ of approximately $0.93399$ reflects the
strong influence of times $T_{k}$ with small $k$ on both $H_{n}$ and $L_{n}$
(Figures 3 and 5). As $n$ increases, $\mathbb{E}\left[I_{n}\right]$ increases
without bound (eq. 16), whereas $\mathbb{E}\left[E_{n}\right]$ remains
constant (eq. 13); the contribution of $E_{n}$ to the total tree length
$L_{n}$ becomes negligible, and $\operatorname{Corr}[L_{n},I_{n}]$ approaches
1. $\operatorname{Corr}[H_{n},I_{n}]$ has the same limiting value as
$\operatorname{Corr}[H_{n},L_{n}]$, and $H_{n}$, $L_{n}$, and $I_{n}$ all have
limiting correlation 0 with $E_{n}$. Interestingly, although $H_{n}$ and
$E_{n}$ have the same limiting expectation of 2, the limit of their
correlation $\operatorname{Corr}[H_{n},E_{n}]$ is 0.

The correlations of $H_{n}$, $L_{n}$, and $I_{n}$ with $B_{n}$, like their
correlations with each other, are relatively high.
$\operatorname{Corr}[H_{n},B_{n}]$ is nearly constant in $n$, with limit
approximately $0.97054$; both $H_{n}$ and $B_{n}$ are determined in large part
by the $T_{k}$ with small $k$ (eqs. 5 and 21), so that little change occurs in
the correlation as $n$ increases. Because $\operatorname{Corr}[H_{n},B_{n}]$
is high and $\operatorname{Corr}[H_{n},L_{n}]$ is also high, the constraint on
a correlation $\operatorname{Corr}[Y,Z]$ given $\operatorname{Corr}[X,Y]$ and
$\operatorname{Corr}[X,Z]$, or (Wickens, , 2014, eq. 7.1),

$\displaystyle\operatorname{Corr}[Y,Z]$ $\displaystyle\geq$
$\displaystyle\operatorname{Corr}[X,Y]\,\operatorname{Corr}[X,Z]-\sqrt{1-\operatorname{Corr}[X,Y]^{2}}\sqrt{1-\operatorname{Corr}[X,Z]^{2}}$
(92) $\displaystyle\operatorname{Corr}[Y,Z]$ $\displaystyle\leq$
$\displaystyle\operatorname{Corr}[X,Y]\,\operatorname{Corr}[X,Z]+\sqrt{1-\operatorname{Corr}[X,Y]^{2}}\sqrt{1-\operatorname{Corr}[X,Z]^{2}},$
(93)

forces a high value for $\operatorname{Corr}[L_{n},B_{n}]$ as well. In
particular, placing $H_{n},L_{n},B_{n}$ in the roles of $X,Y,Z$, with
$\lim_{n\rightarrow\infty}\operatorname{Corr}[H_{n},L_{n}]\approx 0.93399$ and
$\lim_{n\rightarrow\infty}\operatorname{Corr}[H_{n},B_{n}]\approx 0.97054$, we
obtain an interval
$0.82037\leq\lim_{n\rightarrow\infty}\operatorname{Corr}[L_{n},B_{n}]\leq
0.99256$ from eqs. 92 and 93;
$\lim_{n\rightarrow\infty}\operatorname{Corr}[L_{n},B_{n}]\approx 0.84930$
lies near its lower end. Eqs. 92 and 93 similarly force a high value for
$\lim_{n\rightarrow\infty}\operatorname{Corr}[I_{n},B_{n}]$, using
$H_{n},I_{n},B_{n}$ as $X,Y,Z$.

Next, for correlations involving the $T_{k}$, we observe that for fixed $n$,
as $k$ increases from 2 to $n$, $\operatorname{Corr}[H_{n},T_{k}]$ decreases
(Figure 5). At fixed $n$ and $k$, $\operatorname{Corr}[L_{n},T_{k}]$ generally
exceeds $\operatorname{Corr}[H_{n},T_{k}]$; $k$ copies of the branch length
$T_{k}$ contribute to tree length $L_{n}$ (eq. 7), whereas only one copy
contributes to the tree height $H_{n}$ (eq. 4), giving rise to a greater value
for the correlation of $T_{k}$ with $L_{n}$ than with $H_{n}$. For $k>2$,
$\operatorname{Corr}[B_{n},T_{k}]$ is generally smaller than
$\operatorname{Corr}[H_{n},T_{k}]$; because $B_{n}$ is determined to a larger
extent by $T_{2}$ than is $H_{n}$, the correlations of $B_{n}$ with $T_{k}$
for $k>2$ are generally smaller. Finally, because tree length $L_{n}$ consists
primarily of internal branches for large $n$, the correlation
$\operatorname{Corr}[I_{n},T_{k}]$ is similar to
$\operatorname{Corr}[L_{n},T_{k}]$ (Figure 5), approaching the same limit as
$n\rightarrow\infty$ (Table 2); the correlation of $E_{n}$ and $T_{k}$ is a
constant that does not depend on $k$.

##  5 Discussion

We have examined relationships between pairs of tree features under the
coalescent model by deriving expressions for their covariances and correlation
coefficients (Tables 1 and 2). For 13 of 15 pairs examined, we obtained exact
expressions for the covariances and correlation coefficients, and for the
remaining two pairs, we obtained quantities observed in simulations to closely
approximate the desired quantities (Figures 2 and 4). The results provide a
compendium of basic relationships among coalescent tree features, contributing
to a more precise understanding of the way in which the properties of
coalescent trees relate to each other.

In most cases, the covariances have relatively simple expressions, comparable
to the simplicity of most expressions for expectations and variances (Table
1). Expressions for the correlation coefficients are somewhat more complex, in
many cases with $n\rightarrow\infty$ limits that contain terms resulting from
the limit $\sum_{k=1}^{\infty}1/k^{2}=\pi^{2}/6$.

Numerically, we obtain tight correlations between $H_{n}$, $L_{n}$, $I_{n}$,
and $B_{n}$ as $n$ grows large, with all of these quantities possessing
limiting correlations of $0.84930$ or greater (Table 2). In the limit, $L_{n}$
and $I_{n}$ are perfectly correlated, and all limiting correlations of other
quantities with $I_{n}$ are equal to their corresponding correlations with
$L_{n}$. Decreasing correlations are observed for $H_{n}$, $L_{n}$, $I_{n}$,
and $B_{n}$ with $E_{n}$, with limits of 0 observed in all cases (Table 2).
Although $H_{n}$ and $E_{n}$ both have limiting expectation 2 (eqs. 5 and 13),
their limiting correlation coefficient is 0. The correlations among $H_{n}$,
$L_{n}$, and $B_{n}$ are all large; however, the limiting correlation for
$(L_{n},B_{n})$ is near the lower end of the interval suggested by the larger
limiting correlations for $(H_{n},L_{n})$ and $(H_{n},B_{n})$ (eqs. 92 and
93). This result suggests that $L_{n}$ and $B_{n}$ capture relatively distinct
features of coalescent trees in relation to the constraints placed on a pair
of correlated variables that are each highly correlated with a third variable
($H_{n}$). A similar observation can be made concerning $I_{n}$ and $B_{n}$,
as $L_{n}$ and $I_{n}$ are asymptotically fully correlated.

Although tree properties such as $H_{n}$, $L_{n}$, $E_{n}$, $I_{n}$, and
$B_{n}$ are not themselves observable in genetic sequences, interest in these
quantities arises in part from their relationship to statistical tests that
assess the fit of the coalescent model to data on genetic variation. Features
of tree shape underlie predictions of the coalescent regarding allele
frequencies; in particular, tree properties contribute to predictions for the
unfolded site-frequency spectrum (SFS) of a genomic region, the vector that
for a sample of size $n$ tabulates how many variable (biallelic) sites in the
region possess allele frequencies $1/n,2/n,\ldots,(n-1)/n$ for the derived
allele (e.g. Fu, , 1995; Ferretti et al., , 2017). Test statistics then assess
agreement of site-frequency spectra with the predictions (e.g. Zeng et al., ,
2006; Achaz, , 2009; Ferretti et al., , 2010; Ronen et al., , 2013; Fu, ,
2021), so that correlations among statistics emphasizing different aspects of
site-frequency spectra emerge from dependence on correlated tree features. In
this context, further understanding of correlations among tree properties can
assist in understanding the joint behavior of SFS-based tests of the
coalescent model.

Our computations augment earlier calculations concerning quantities associated
with coalescent trees. The pairs $(H_{n},L_{n})$ (Arbisser et al., , 2018) and
$(L_{n},E_{n})$ and $(E_{n},I_{n})$ (Fu & Li, , 1993) have been studied in
detail. Results for pairs $(H_{n},T_{k})$, $(L_{n},T_{k})$, and
$(L_{n},I_{n})$ follow trivially from the derivations and results of Arbisser
et al.,  (2018) and Fu & Li,  (1993), but were not highlighted in those
studies. Results for pairs $(H_{n},E_{n})$, $(H_{n},I_{n})$, $(E_{n},T_{k})$,
and $(I_{n},T_{k})$ follow from derivations similar to those of Fu & Li,
(1993), but to our knowledge, they have not been previously reported.

The least-studied of the variables we consider, $B_{n}$, was introduced by
Uyenoyama,  (1997) in the context of balancing selection and self-
incompatibility alleles in plants. Under balancing selection, the mean $B_{n}$
of the two basal branches is expected to be long in relation to the tree
length $L_{n}$, so that $2B_{n}/L_{n}$ predicts the fraction of segregating
sites that distinguish two long-separated sets of lineages. For $B_{n}$, which
gives a portion of the height $H_{n}$—but which, unlike $H_{n}$, is obtained
from a sum with a random length—we derived the variance (eq. 22), as well as
exact covariances and correlations with $H_{n}$, $L_{n}$, and $T_{k}$ and
approximate covariances and correlations with $E_{n}$ and $I_{n}$. Several
studies have extended the work of Fu & Li,  (1993) on features of the external
and internal branch lengths (Blum & François, , 2005; Caliebe et al., , 2007;
Janson & Kersting, , 2011; Dahmer & Kersting, , 2015, 2017; Disanto & Wiehe, ,
2020); it may be possible to seek exact rather than approximate covariances
and correlations for $(E_{n},B_{n})$ and $(I_{n},B_{n})$ by building on these
studies.

When examining joint distributions of $H_{n}$ and $L_{n}$, Arbisser et al.,
(2018) used computations of the expectations and variances of $H_{n}$ and
$L_{n}$ and the covariance of $H_{n}$ and $L_{n}$ to obtain approximations for
the expectation and variance of $H_{n}/L_{n}$. Following the approach of
Arbisser et al.,  (2018), our results could be used to obtain similar
approximate expressions for expectations and variances of ratios of additional
pairs.

Acknowledgments. We thank Yun-Xin Fu and John Wakeley for comments on the
manuscript. We acknowledge support from NIH grants R01 GM131404 and R01
HG005855 and NSF grant BCS-2116322.

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$(X_{n},Y_{n})$ | $\operatorname{Cov}[X_{n},Y_{n}]$ | $\lim_{n\rightarrow\infty}\operatorname{Cov}[X_{n},Y_{n}]$ |  Reference
---|---|---|---
$H_{n}$, $T_{k}$ | $\frac{4}{k^{2}(k-1)^{2}}$ | $\frac{4}{k^{2}(k-1)^{2}}$ |  3.1, eq. 23
$H_{n}$, $L_{n}$ | $4S_{2,n-1}-4+\frac{4}{n}$ | $\frac{2\pi^{2}}{3}-4\approx 2.57974$ |  3.3, eqs. 29, 30
$H_{n}$, $E_{n}$ | $\frac{4}{n}$ | 0 |  3.4, eqs. 37, 38
$H_{n}$, $I_{n}$ | $4S_{2,n-1}-4$ | $\frac{2\pi^{2}}{3}-4\approx 2.57974$ |  3.7, eqs. 49, 50
$H_{n}$, $B_{n}$ | $\frac{4[S_{3,n-1}n^{2}-3S_{2,n-1}n^{2}+(n-1)(4n+1)]}{n^{2}}$ | $4\zeta(3)+16-2\pi^{2}\approx 1.06902$ |  3.13, eqs. 71, 72
$L_{n}$, $T_{k}$ | $\frac{4}{k(k-1)^{2}}$ | $\frac{4}{k(k-1)^{2}}$ |  3.2, eq. 26
$L_{n}$, $E_{n}$ | $\frac{4S_{1,n-1}}{n-1}$ | 0 |  3.6, eqs. 45, 46
$L_{n}$, $I_{n}$ | $4S_{2,n-1}-\frac{4S_{1,n-1}}{n-1}$ | $\frac{2\pi^{2}}{3}\approx 6.57974$ |  3.9, eqs. 57, 58
$L_{n}$, $B_{n}$ | $\frac{4[S_{3,n-1}n-S_{2,n-1}n+n-1]}{n}$ | $4\zeta(3)+4-\frac{2\pi^{2}}{3}\approx 2.22849$ |  3.14, eqs. 75, 76
$E_{n}$, $T_{k}$ | $\frac{4}{k(k-1)(n-1)}$ | 0 |  3.5, eqs. 41, 42
$E_{n}$, $I_{n}$ | $\frac{4S_{1,n-1}}{n-1}-\frac{8S_{1,n-1}n}{(n-1)(n-2)}+\frac{16}{n-2}$ | 0 |  3.10, eqs. 61, 62
$E_{n}$, $B_{n}$ | $\frac{4\left(S_{2,n-1}n-n+1\right)}{n(n-1)}$ | 0 |  3.15, eqs. 84, 85
$I_{n}$, $T_{k}$ | $\frac{4(n-k)}{k(k-1)^{2}(n-1)}$ | $\frac{4}{k(k-1)^{2}}$ |  3.8, eqs. 53, 54
$I_{n}$, $B_{n}$ | $\frac{4(S_{3,n-1}n-S_{2,n-1}n+n-S_{3,n-1}-1)}{n-1}$ | $4\zeta(3)+4-\frac{2\pi^{2}}{3}\approx 2.22849$ |  3.16, eqs. 88, 89
$B_{n}$, $T_{k}$ | $\frac{4}{k^{2}(k-1)^{3}}$ | $\frac{4}{k^{2}(k-1)^{3}}$ |  3.12, eq. 68
Table 1: Covariances of pairs of variables. Expressions involving $E_{n}$ or
$I_{n}$ apply for $n\geq 3$ and expressions involving $B_{n}$ apply for $n\geq
4$.

$(X_{n},Y_{n})$ | $\operatorname{Corr}[X_{n},Y_{n}]$ | $\lim_{n\rightarrow\infty}\operatorname{Corr}[X_{n},Y_{n}]$ |  Reference
---|---|---|---
$H_{n}$, $T_{k}$ | $\frac{n}{\sqrt{2n^{2}S_{2,n}-3n^{2}+2n-1}}\frac{1}{k(k-1)}$ | $\frac{\sqrt{3}}{\sqrt{\pi^{2}-9}}\frac{1}{k(k-1)}\approx\frac{1.85738}{k(k-1)}$ |  3.1, eqs. 24, 25
$H_{n}$, $L_{n}$ | $\frac{S_{2,n-1}n-n+1}{\sqrt{S_{2,n-1}\left(2S_{2,n}n^{2}-3n^{2}+2n-1\right)}}$ | $\frac{\pi^{2}-6}{\pi\sqrt{2(\pi^{2}-9)}}\approx 0.93399$ |  3.3, eqs. 31, 32
$H_{n}$, $E_{n}$ | $\frac{\sqrt{(n-1)(n-2)}}{\sqrt{2(2S_{2,n}n^{2}-3n^{2}+2n-1)\left(S_{1,n-1}n-2n+2\right)}}$ | 0 |  3.4, eqs. 39, 40
$H_{n}$, $I_{n}$ | $\frac{\left(S_{2,n-1}-1\right)n\sqrt{(n-1)(n-2)}}{\left(2S_{2,n}n^{2}-3n^{2}+2n-1\right)\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}$ | $\frac{\pi^{2}-6}{\pi\sqrt{2(\pi^{2}-9)}}\approx 0.93399$ |  3.7, eqs. 51, 52
$H_{n}$, $B_{n}$ | $\frac{\sqrt{2}[S_{3,n-1}n^{2}-3S_{2,n-1}n^{2}+(n-1)(4n+1)]}{\sqrt{(2S_{2,n}n^{2}-3n^{2}+2n-1)(3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4})}$ | $\frac{3\sqrt{3}[2\zeta(3)+8-\pi^{2}]}{\sqrt{(\pi^{2}-9)(18+9\pi^{2}-\pi^{4})}}\approx 0.97054$ |  3.13, eqs. 73, 74
$L_{n}$, $T_{k}$ | $\frac{1}{\sqrt{S_{2,n-1}}}\frac{1}{k-1}$ | $\frac{\sqrt{6}}{\pi}\frac{1}{k-1}\approx\frac{0.77970}{k-1}$ |  3.2, eqs. 27, 28
$L_{n}$, $E_{n}$ | $\frac{S_{1,n-1}\sqrt{n-2}}{\sqrt{2S_{2,n-1}(n-1)\left(S_{1,n-1}n-2n+2\right)}}$ | 0 |  3.6, eqs. 47, 48
$L_{n}$, $I_{n}$ | $\frac{[S_{2,n-1}(n-1)-S_{1,n-1}]\sqrt{n-2}}{(n-1)\sqrt{S_{2,n-1}\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}}$ | 1 |  3.9, eqs. 59, 60
$L_{n}$, $B_{n}$ | $\frac{\sqrt{2}(S_{3,n-1}n-S_{2,n-1}n+n-1)}{\sqrt{S_{2,n-1}(3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4})}$ | $\frac{\sqrt{6}[6\zeta(3)+6-\pi^{2}]}{\pi\sqrt{18+9\pi^{2}-\pi^{4}}}\approx 0.84930$ |  3.14, eqs. 77, 78
$E_{n}$, $T_{k}$ | $\frac{\sqrt{n-2}}{\sqrt{2(n-1)(S_{1,n-1}n-2n+2)}}$ | 0 |  3.5, eqs. 43, 44
$E_{n}$, $I_{n}$ | $\frac{4(n-1)-S_{1,n-1}(n+2)}{\sqrt{2\left(S_{1,n-1}n-2n+2\right)\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}}$ | 0 |  3.10, eqs. 63, 64
$E_{n}$, $B_{n}$ | $\frac{(S_{2,n-1}n-n+1)\sqrt{n-2}}{\sqrt{(n-1)(S_{1,n-1}n-2n+2)(3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4)}}$ | 0 |  3.15, eqs. 86, 87
$I_{n}$, $T_{k}$ | $\frac{(n-k)\sqrt{n-2}}{(k-1)\sqrt{(n-1)\left[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)\right]}}$ | $\frac{\sqrt{6}}{\pi}\frac{1}{k-1}\approx\frac{0.77970}{k-1}$ |  3.8, eqs. 55, 56
$I_{n}$, $B_{n}$ | $\frac{\sqrt{2}(S_{3,n-1}n-S_{2,n-1}n+n-S_{3,n-1}-1)n\sqrt{n-2}}{\sqrt{(n-1)[4S_{1,n-1}+S_{2,n-1}(n-1)(n-2)-4(n-1)](3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4)}}$ | $\frac{\sqrt{6}[6\zeta(3)+6-\pi^{2}]}{\pi\sqrt{18+9\pi^{2}-\pi^{4}}}\approx 0.84930$ |  3.16, eqs. 90, 91
$B_{n}$, $T_{k}$ | $\frac{\sqrt{2}n}{\sqrt{3S_{2,n-1}n^{2}-2S_{2,n-1}^{2}n^{2}+n^{2}-4S_{2,n-1}n+3n-4}}\frac{1}{k(k-1)^{2}}$ | $\frac{6}{\sqrt{18+9\pi^{2}-\pi^{4}}}\frac{1}{k(k-1)^{2}}\approx\frac{1.95518}{k(k-1)^{2}}$ |  3.12, eqs. 69, 70
Table 2: Correlation coefficients of pairs of variables. Expressions involving
$E_{n}$ or $I_{n}$ apply for $n\geq 3$ and expressions involving $B_{n}$ apply
for $n\geq 4$.  Figure 1: Tree features. The tree height is $H_{n}$ and the
sum of the lengths of all tree branches is $L_{n}$. External branches, with
total length $E_{n}$, appear in green; internal branches, with total length
$I_{n}$, appear in red; basal branches, with mean length $B_{n}$, appear in
purple.

Figure 2: Simulated and theoretical values of covariances of pairs of
variables, plotted as functions of sample size $n$. Expressions for
theoretical values are taken from Table 1.

Figure 3: Theoretical values of covariances $\operatorname{Cov}[X,T_{k}]$ for
variables $X$ in $\\{H_{n},L_{n},E_{n},I_{n},B_{n}\\}$, plotted as functions
of $k$ for $n=10$, $n=20$, and $n=50$. The plots appear on a logarithmic
scale.

Figure 4: Simulated and theoretical values of correlation coefficients of
pairs of variables, plotted as functions of sample size $n$. Expressions for
theoretical values are taken from Table 2.

Figure 5: Theoretical values of correlation coefficients
$\operatorname{Corr}[X,T_{k}]$ for variables $X$ in
$\\{H_{n},L_{n},E_{n},I_{n},B_{n}\\}$, plotted as functions of $k$ for $n=10$,
$n=20$, and $n=50$.