Abstract Given a tree T and a vertex v in T , the number of subtrees of T containing v is denoted by η T ( v ) and called the subtree number at v . Similarly, η T ( u , v ) denotes the number of subtrees of T that contain both u and v . Motivated from the distance function and the eccentricity at a vertex, we study the eccentric subtree number at a vertex v in T , defined as η T e c c ( v ) = min u ∈ V ( T ) η T ( v , u ) . This concept provides us a new perspective in the study of the well-known correlation between the distance and the subtree numbers. We will present properties of the eccentric subtree number analogous to those related to the eccentricity and center of a tree, as well as some extremal results with respect to the eccentric subtree number and sum of eccentric subtree numbers in a tree. Some open problems for further study are also proposed.

中文翻译：

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关于树中偏心子树数

摘要 给定一棵树 T 和 T 中的一个顶点 v ，T 中包含 v 的子树数用 η T ( v ) 表示，称为 v 处的子树数。类似地， η T ( u , v ) 表示包含 u 和 v 的 T 子树的数量。受距离函数和顶点偏心度的启发，我们研究了 T 中顶点 v 处的偏心子树数，定义为 η T ecc ( v ) = min u ∈ V ( T ) η T ( v , u ) 。这个概念为我们研究距离和子树数之间众所周知的相关性提供了一个新的视角。我们将展示偏心子树数的性质，类似于与树的偏心度和中心相关的性质，以及关于偏心子树数和树中偏心子树数之和的一些极值结果。