Centroid, Leaf-Centroid, and Internal-Centroid

We consider some different distance-based “central parts” of a tree including sets of vertices that minimize the sum of distances to: all other vertices (the centroid of a tree), all leaves (the leaf-centroid of a tree), all internal vertices (the internal-centroid of a tree). The subgraphs induced by these “central parts” are briefly discussed. Regarding their relative locations in the same tree, it is shown that the centroid is always located in the “middle” of the leaf-centroid and internal-centroid. In a tree T of order n, the distance between the leaf-centroid and the centroid or the internal centroid can be as large as n/2 (asymptotically); the distance between the internal centroid and the centroid, however, can only be as large as n/4 (asymptotically). All extremal cases are obtained by the so called comets. We also point out that this study can be further generalized to trees with additional constraints on the diameter or vertex degrees. The arguments are very similar but with more technical calculations.