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