Robust Common Edges of Two Random Spanning Trees

Let $T_1$ and $T_2$ be two independent uniformly random spanning trees of the complete graph $K_n$ (each drawn uniformly at random from all $n^{n-2}$ labeled spanning trees, independently of the other). Find the expected number of edges that belong to both $T_1$ and $T_2$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.