Metric trees of generalized roundness one

TL;DR

This paper characterizes classes of countable metric trees with generalized roundness exactly one, including spherically symmetric trees with infinite branching and other non-symmetric trees with specific spreading or comb-like structures.

Contribution

It identifies large classes of countable metric trees with generalized roundness one, extending previous results beyond finite trees and symmetric cases.

Findings

Countable spherically symmetric trees with infinite branching have generalized roundness one.

Certain non-symmetric trees with sufficient spreading also have generalized roundness one.

Provides bounds and conditions based on degree sequences and tree structure.

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 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 $(d_{0}, d_{1}, d_{2}...)$ 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. 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.