Tutte polynomials for trees

We define two two‐variable polynomials for rooted trees and one two‐variable polynomial for unrooted trees, all of which are based on the coranknullity formulation of the Tutte polynomial of a graph or matroid. For the rooted polynomials, we show that the polynomial completely determines the rooted tree, i.e., rooted trees T 1 and T 2 are isomorphic if and only if f ( T 1 ) = f ( T 2 ). The corresponding question is open in the unrooted case, although we can reconstruct the degree sequence, number of subtrees of size k for all k , and the number of paths of length k for all k from the (unrooted) polynomial. The key difference between these three polynomials and the standard Tutte polynomial is the rank function used; we use pruning and branching ranks to define the polynomials. We also give a subtree expansion of the polynomials and a deletion‐contraction recursion they satisfy.