I wonder what is known on the following:

1) What is the number $T_k(n)$ of $k$-tuples of (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ with $n$ labelled vertices?

2) (harder, it seems) What is the number $S_k(n)$ of subgraphs with $n$ labelled vertices which are union of $k$ (pairwise) edge-disjoint trees?

3) What is the number $R_k(n)$ of $k$-tuples of (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ on the set of vertices $[n]={1,2,\dots,n}$ such that $T_i$ is a tree with $n-i$ edges?

4) (Like 2) What is the number $U_k(n)$ of subgraphs with $n$ labelled vertices which are union of $k$ (pairwise) edge-disjoint trees $(T_1,T_2,\dots, T_k)$ on the set of vertices $[n]={1,2,\dots,n}$ such that $T_i$ is a tree with $n-i$ edges?

(Added Sept 7, 2017): 5) What is the number $F_k(n)$ of $k$-tuples of (pairwise) edge-disjoint forests $(F_1,F_2,\dots, F_k)$ on the set of vertices $[n]={1,2,\dots,n}$ such that $F_i$ is a spanning forest with $n-i$ edges?

6) Variant on 5): Rooted forests rather than forests. (Call the number $G_k(n)$.)

Of course, Cayley's formula for the number of trees on $n$ labelled vertices gives that $T_1(n)=S_1(n)=R_1(n)=U_1(n)=F_1(n)=G_1(n)=n^{n-2}$ but I wonder if something is known even for $k=2$.