  Consider a tree $T=(V,E)$ with root $\circ$ and edge length function
$\ell:E\to\mathbb{R}_+$. The phylogenetic covariance matrix of $T$ is the
matrix $C$ with rows and columns indexed by $L$, the leaf set of $T$, with
entries $C(i,j):=\sum_{e\in[i\wedge j,o]}\ell(e)$, for each $i,j\in L$. Recent
work [15] has shown that the phylogenetic covariance matrix of a large, random
binary tree $T$ is significantly sparsified with overwhelmingly high
probability under a change-of-basis with respect to the so-called Haar-like
wavelets of $T$. This finding notably enables manipulating the spectrum of
covariance matrices of large binary trees without the necessity to store them
in computer memory but instead performing two post-order traversals of the
tree. Building on the methods of [15], this manuscript further advances their
sparsification result to encompass the broader class of $k$-regular trees, for
any given $k\ge2$. This extension is achieved by refining existing asymptotic
formulas for the mean and variance of the internal path length of random
$k$-regular trees, utilizing hypergeometric function properties and identities.
