For a local field $K$ and $n \geq 2$, let $��_n$ and $��_n$ denote the affine buildings naturally associated to the special linear and symplectic groups $\SL_n(K)$ and $\Sp_n(K)$, respectively. We relate the number of vertices in $��_n$ ($n \geq 3$) close (i.e., gallery distance 1) to a given vertex in $��_n$ to the number of chambers in $��_n$ containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in $��_n$ ($n \geq 2$) close to a given special vertex in $��_n$ (all the vertices in $��_n$ are special) and establish analogues of our results for $��_n$.

16 pages, 3 figures; minor corrections; accepted for publication in INTEGERS: The Electronic Journal of Combinatorial Number Theory