Maximum centers of caterpillars

Characterize the locations of maximum r-centers in caterpillars.

Background

The survey discusses several results on maximum centers in trees and states a separate conjecture concerning the degree of a neighbor of a maximum center in a caterpillar. This question asks for a complete structural description rather than only a necessary degree condition.

References

Where are the maximum $r$-centers for caterpillars?

— A Survey of the Holroyd-Talbot Conjecture  (2501.16144 - Hurlbert, 27 Jan 2025) in Question 2(b), Section 6