An area-depth symmetric $q,t$-Catalan polynomial

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also answer an open question of Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.