Non-isomorphic caterpillars with identical subtree data

The greedoid Tutte polynomial of a tree is equivalent to a generating function that encodes information about the number of subtrees with I internal (non-leaf) edges and L leaf edges, for all I and L. We prove that this information does not uniquely determine the tree T by constructing an infinite family of pairs of non-isomorphic caterpillars, each pair having identical subtree data. This disproves conjectures of [S. Chaudhary, G. Gordon, Tutte polynomials for trees, J. Graph Theory 15 (1991) 317-331] and [G. Gordon, E. McDonnell, D. Orloff, N. Yung, On the Tutte polynomial of a tree, Congr. Numer. 108 (1995) 141-151] and contrasts with the situation for rooted trees, where this data completely determines the rooted tree.