We define two two-variable polynomials for rooted trees and one twovariable 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 TI and T, are isomorphic if and only if f(T,) = f(T2). 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.

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