The number of subtrees, also referred to as the subtrees index, is a key parameter to measure graph structures such as networks. In this paper, we investigate the number of subtrees of planar two-tree networks. By “adding a virtual edge” and “edge orientation”, we present a linear time algorithm for computing the number of subtrees of planar two-tree networks, as well as a family of planar two-connected networks. As applications, we provide the formulae for the number of subtrees of the famous small-world Farey network and GDURT network. We also discuss the relationship between the spanning subtree number and the subtree number of these networks.

中文翻译：

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平面二树网络子树的枚举

子树的数量，也称为子树索引，是衡量网络等图结构的关键参数。在本文中，我们研究了平面二树网络的子树数量。通过“添加虚拟边”和“边方向”，我们提出了一种线性时间算法，用于计算平面二树网络的子树数量，以及平面二连通网络族。作为应用，我们提供了著名的小世界 Farey 网络和 GDURT 网络的子树数公式。我们还讨论了这些网络的生成子树数和子树数之间的关系。