We study weighted periodic-orbit zeta functions for an infinitely cusped tree lattice $Y_q$, where $q\ge2$ is even and the quotient is a one-sided comb. The global Euler product fails coefficientwise because infinitely many primitive cycles have length four. A first-return determinant at a finite directed-edge set nevertheless exists, and stationary Schur elimination gives an algebraic formula for the root local zeta and its dominant poles. For the two-step multiplicity potential we compute the Gurevich pressure $P_G=2\log(q+1)$ and pressure at infinity $P_\infty=\log(4q)$, yielding strong positive recurrence and exponential local-orbit asymptotics. The height-damped transition operator is trace class. After subtraction of an explicit integrated-pressure counterterm, the inverse Fredholm determinant has a locally uniform finite part, expressed by a convergent dilogarithmic product and covariant under changes of height.
We study the standard nonuniform arithmetic quotient of the affine Bruhat--Tits building attached to $\operatorname{PGL}_d(\mathbb F_q(\!(t^{-1})\!))$, with Haar measure normalized so that a maximal compact subgroup has volume one. We first compute its vertex volume in closed product form. The proof is entirely building-theoretic: vertices are parametrized by a dominant sector, their stabilizers are counted exactly, and the resulting sum over block compositions is evaluated by a cut-set recursion. On the same quotient, we introduce a homothety-invariant normalized lattice-minima height $\alpha$. We determine its exact integrability threshold, proving that $\alpha$ belongs to $L^r$ precisely for $0<r<d$, and establish a sharp cusp-tail estimate of order $T^{-d}$. The associated positive-moment height zeta function, equivalently the Mellin transform of the cusp-height distribution, converges exactly in the half-plane $\operatorname{Re}(s)<d$. It admits a meromorphic continuation as a rational function of $q^{s/d}$ and has a simple pole at $s=d$, with an explicit critical coefficient. We also compute the resulting rational functions explicitly for $d=3,4,5$. Thus the same dominant-sector coordinates simultaneously control volume, cusp decay, and the analytic structure of the height zeta function.
