Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify several large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual path metric (SSTs). Using a simple geometric argument we show how to determine reasonable 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 (d0, d1, d2, . . .) of an SST (T, ρ) satisfies |{j | dj > 1}| = א0, then (T, ρ) has generalized roundness one. In particular, all complete n-ary trees of depth ∞ (n ≥ 2), all k-regular trees (k ≥ 3) and all inductive limits of Cantor trees are seen to have generalized roundness one. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs. It remains an intriguing problem to completely classify countable metric trees of generalized roundness one. Mathematics Subject Classification (2010). Primary 46B85; Secondary 54E35.