In this paper we consider spherically symmetric trees endowed with the usual
combinatorial metric (SSTs). Using a simple geometric argument we show how to
determine decent upper bounds on the generalized roundness of finite SSTs that
depend only on the downward degree sequence of the tree in question. By
considering limits it follows that if the downward degree sequence $(d_{0},
d_{1}, d_{2}, \ldots)$ of a SST $(T,\rho)$ satisfies $|\{ j \, | \, d_{j} > 1
\}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included
among the trees that satisfy this condition are all complete $n$-ary trees of
depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive
limits of Cantor trees.