Metric trees of generalized roundness one

Abstract

Abstract. 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 some 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 sequencehas 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.

Comparing the Generalised Roundness of Metric Spaces

“…If f (k) = 1 for all k ≥ 0, the resulting metric trees C( f ) and C m ( f ) will be denoted by C(1) and C m (1), respectively. In other words, C(1) and C m (1) are the combs C and C m endowed with the usual combinatorial path metric δ. Caffarelli et al [2] have shown that ℘ C m (1) → 1 as m → ∞. Hence, ℘ C(1) = 1.…”

“…We proceed to show that large classes of divergent and convergent SSTs have generalised roundness one. The following lemma is a variation of [2,Theorem 2.1]. As the statement of the lemma is complicated, we will comment on the intuition behind this result.…”

“…Caffarelli et al [2] considered the generalised roundness of spherically symmetric trees endowed with the usual combinatorial path metric (wherein all edges in the tree are assumed to have unit length). In this section we consider a broader class of spherically symmetric trees by relaxing the requirement that all edges have unit length.…”

“…This fact is folklore and it may be derived in several different ways (see Kelly [21,Theorem II] and Faver et al [10,Proposition 4.1]). Examples of Caffarelli et al [2] show that some countable metric trees have generalised roundness exactly one. The situation is different for finite metric trees.…”

“…Caffarelli et al [2] identified several classes of metric trees of generalised roundness one. The types of trees studied in [2] were spherically symmetric, infinitely bifurcating or comb-like trees endowed with the usual combinatorial path metric. In other words, all edges in the trees were assumed to have length one and all other distances were determined geodesically.…”