Spiders are status unique in trees

The status of a vertex xx in a graph is the sum of the distances between xx and all other vertices. The status sequence of a graph is the list of the statuses of all vertices arranged in nondecreasing order. It is well known that non-isomorphic trees may have the same status sequence. A graph GG is said to be status unique in a family FF of graphs if GG is a member of FF and GG is uniquely determined in FF by its status sequence. The main result of this paper is that every spider is status unique in the family of all trees. Some conjectures about status unique graphs are proposed at the end of the paper.