Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials

We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schr\"oder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.
Comment: LaTeX2e, 36 pages (includes 4 figures). Version 2 corrects a small error in the definition of crossing number (p. 6) and includes a proof of the previously conjectured (1.25)/(1.26)