Maximum possible order of a tree betweenness center

Determine the maximum value f(n) of the order of the betweenness center among all trees of order n, or at least determine the asymptotic order of growth of f(n), and establish whether f(n)=o(n).

Background

The paper proves that the betweenness center of every tree is contained in a path and provides a census of trees up to order 20 classified by the order of their betweenness centers. Although trees with arbitrarily large betweenness centers are known to exist, the enumerated examples have relatively small centers, motivating the definition of f(n) as the largest possible betweenness-center order for an n-vertex tree.

The unresolved problem asks for an exact determination or asymptotic characterization of f(n), including the specific question of whether the maximum center size is sublinear in the tree order.