Erdős-Ko-Rado theorems for a family of trees

Hurlbert and Kamat had conjectured that one can always find a largest r-star of a tree whose centre is a leaf. Baber and Borg have independently shown this to be false. We show that, moreover, for all integers n≥2 and d≥3, there exists a positive integer r such that there is a tree where the centre of the largest r-star is a vertex of degree n at distance d from every leaf.