A complete $t$-intersection theorem for families of spanning trees

Let ${\mathcal{T}}_n$ denote the set of all labelled spanning trees of $K_n$. A family $\mathcal{F}\subset {\mathcal{T}}_n$ is $t$-intersecting if for all $A,B\in \mathcal{F}$ the trees $A$ and $B$ share at least $t$ edges. In this paper, we determine for $n>n_0$ the size of the largest $t$-intersecting family $\mathcal{F}\subset {\mathcal{T}}_n$ for all meaningful values of $t$ ($t\le n-1$). This result is a rare instance when a complete $t$-intersection theorem for a given type of structures is known.