A fundamental problem in network science is the normalization of the topological or physical distance between vertices, which requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems of the sum of edge lengths in trees of a fixed size: the minimum and the maximum value of the sum for specific trees, the minimum and the maximum in classes of trees (bistar trees and caterpillar trees) and finally the minimum and the maximum for any tree. We establish some foundations for research on optimality scores for spatial networks in one dimension.

网络科学中的一个基本问题是顶点之间的拓扑或物理距离的归一化，这需要了解未归一化距离的变化范围。在这里，我们研究了树木顶点线性排列中物理距离变化的极限。特别是，我们研究了固定大小的树中边长之和的各种问题：特定树的总和的最小值和最大值，树木类别（双星树和毛毛虫树）的最小值和最大值，以及最后是任何树的最小值和最大值。我们为一维空间网络的最优得分研究奠定了基础。