  We study the group $C^*$-algebras $C^*_{L^{p+}}(G)$ - constructed from
$L^p$-integrability properties of matrix coefficients of unitary
representations - of locally compact groups $G$ acting on (semi-)homogeneous
trees of sufficiently large degree. These group $C^*$-algebras lie between the
universal and the reduced group $C^*$-algebra. By directly investigating these
$L^p$-integrability properties, we first show that for every non-compact,
closed subgroup $G$ of the automorphism group $\mathrm{Aut}(T)$ of a
(semi-)homogeneous tree $T$ that acts transitively on the boundary $\partial T$
and every $2 \leq q < p \leq \infty$, the canonical quotient map
$C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ is not injective. This
reproves a result of Samei and Wiersma. We prove that under the additional
assumptions that $G$ acts transitively on $T$ and that it has Tits'
independence property, the group $C^*$-algebras $C^*_{L^{p+}}(G)$ are the only
group $C^*$-algebras coming from $G$-invariant ideals in the Fourier-Stieltjes
algebra $B(G)$. Additionally, we show that given a group $G$ as before, every
group $C^*$-algebra $C^*_{\mu}(G)$ that is distinguishable (as a group
$C^*$-algebra) from the universal group $C^*$-algebra of $G$ and whose dual
space $C^*_\mu(G)^*$ is a $G$-invariant ideal in $B(G)$ is abstractly
${}^*$-isomorphic to the reduced group $C^*$-algebra of $G$.
