How to count trees?

We propose a new topological invariant of unlabeled trees of N nodes. The invariant is a set of Nx2 matrices of integers, with sum_j k^{d_{i,j}} and v_i as the matrix elements, where d_{i,j} are the elements of the distance matrix and v_i denotes i-th node's degree and k in N. To compare the invariant calculated for possibly different graphs, the matrix rows are ordered with respect to first column, and -- if necessary -- with respect to the second one. We use the new invariant to evaluate from below the number of topologically different unlabeled trees up to N=17. The results slightly exceed the asymptotic evaluation of Otter.