On the eccentric subtree number in trees

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^ecc(v)=minu∈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.