  In this paper we exhibit two infinite families of trees $\{T^1_n\}_{n \geq
17}$ and $\{T^2_n\}_{n \geq 17}$ on $n$ vertices, such that $T^1_n$ and $T^2_n$
are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs)
based on $T^1_n$ and $T^2_n$ have the same geodesic growth with respect to the
standard generating set. We then show that the spectrum of a tree does is not
sufficient to determine the geodesic growth of the RACG based on that tree, by
providing two infinite families of trees $\{S^1_n\}_{n \geq 11}$ and
$\{S^2_n\}_{n \geq 11}$, on $n$ vertices, such that $S^1_n$ and $S^2_n$ are
non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based
on $S^1_n$ and $S^2_n$ have distinct geodesic growth.
  Asymptotically, as $n\rightarrow \infty$, each set $T^i_n$, or $S^i_n$,
$i=1,2$, has the cardinality of the set of all trees on $n$ vertices. Our
proofs are constructive and use two families of trees previously studied by B.
McKay and C. Godsil.
