广义圆度1的度量树
Metric trees of generalized roundness one
摘要：
每个有限度量树的广义圆度严格大于1。另一方面，一些可数度量树的广义圆度恰好为1。本文的目的是确定几大类可数度量树的广义圆度恰好为1。首先，我们考虑球对称树赋予通常的组合度量（SST）。使用一个简单的几何参数，我们展示了如何确定体面的上界的广义圆的有限SST，只依赖于向下度序列的树的问题。通过考虑极限，得出如果SST $(T,ρ)$的向下度序列$(d_{0}, d_{1}, d_{2}...)$满足$|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$，则$(T,ρ)$具有广义圆度1。满足这个条件的树包括深度为$\infty$（$n \geq 2$）的所有完全$n$ -ary树、所有$k$ -正则树（$k \geq 3$）和康托树的归纳极限.其余的文件涉及两类可数度量树的广义圆度的成员之一，在一般情况下，球对称。第一类这样的树只需要以足够的速度展开（限制叶子的数量），第二类这样的树类似于无限的梳子。
英文摘要：
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 $(T,ρ)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,ρ)$ 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.
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