Joining caterpillars and stability of the tree center

The path-table  P(T)P(T) of a tree T collects information regarding the paths in T  : for each vertex vv, the row of P(T)P(T) relative to vv lists the number of paths containing vv of the various lengths. We call this row the path-row   of vv in T.Two trees having the same path-table (up to reordering the rows) are called path-congruent (or path-isomorphic).Motivated by Kelly–Ulam's Reconstruction Conjecture and its variants, we have looked for new necessary and sufficient conditions for isomorphisms between two trees.Path-congruent trees need not be isomorphic, although they are similar in some respects. In [P. Dulio, V. Pannone, Trees with path-stable center, Ars Combinatoria, LXXX (2006) 153–175] we have introduced the concepts of trunk  Tr(T)Tr(T) of a tree T and ramification  ramv of a vertex v∈V(Tr(T))v∈V(Tr(T)), and proved that, if the ramification of the central vertices attains its minimum or maximum value, then the path-row of a central vertex is “unique”, i.e. it is different from the path-row of any non-central vertex (in fact, this uniqueness property of a central path-row holds for all trees of diameter less than 8, regardless of the ramification values).In this paper we prove that, for all other values of the ramification, and for all diameters greater than 7, there are trees in which the above uniqueness fails.