Maximum centers in caterpillars

Characterize the maximum r-centers of caterpillar graphs for every admissible integer r.

Background

The survey notes that caterpillars are known to have the stronger minimum-degree-center property, but it does not provide a complete description of which vertices are maximum r-centers.

The question therefore asks for a structural characterization of all maximum centers in caterpillars, rather than merely proving existence of one suitable center.