On extremal leaf status and internal status of trees

Abstract: For a vertex $u$ of a tree $T$, the leaf (internal, respectively) status of $u$ is the sum of the distances from $u$ to all leaves (internal vertices, respectively) of $T$. The minimum (maximum, respectively) leaf status of a tree $T$ is the minimum (maximum, respectively) leaf statuses of all vertices of $T$. The minimum (maximum, respectively) internal status of a tree $T$ is the minimum (maximum, respectively) internal statuses of all vertices of $T$. We give the smallest and largest values for the minimum leaf status, maximum leaf status, minimum internal status, and maximum internal status of a tree and characterize the extremal cases. We also discuss these parameters of a tree with given diameter or maximum degree.