Spiders are status unique in trees

The status of a vertex x in a graph is the sum of the distances between x 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 G is said to be status unique in a family F of graphs if G is a member of F and G is uniquely determined in F 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.