Two $t$-analogues of the tree inversion enumerator

In this note, we introduce two $t$-analogues $I_n(q,t)$ and ${\tilde{I}}_n(q,t)$ of the tree inversion enumerator $I_n(q)$. Although similar, $I_n(q,t)$ and ${\tilde{I}}_n(q,t)$ are different. But they both seem to have interesting properties. In particular, we conjecture that their $q=-1$ specializations give two different, natural refinements of the zigzag numbers counting alternating permutations.