The bipartition polynomial \(B(G,u,v,w)\) associated with a finite simple undirected graph G has been defined by Dod, Kotek, Preen and Tittmann. It is a common generalization of the domination polynomial, the Ising polynomial and a few others. These authors conjecture that for a finite tree T, \(B(T,u,v,w)\) determines T up to isomorphism. We prove a weak form of the conjecture for caterpillars and we propose a method for extending the proof to rooted trees. A caterpillar is a path with additional pendent edges. For that, we introduce an alternative polynomial \(Q(G,u,v,w)\) that is symmetric in the sense that \(Q(G,u,v,w)=Q(G,v,u,w)\) and is equivalent to \(B(G,u,v,w)\) (up to algebraic transformations). We can thus work on the conjecture with Q instead of B. This polynomial can be computed by a finite automaton on binary terms denoting finite rooted trees.