I recently watched the Numberphile video on the number TREE(3) and where it comes from. At the end, they present a possible starting sequence shown here.
To recap, here are the two rules for building the forest:
- The nth tree must contain at maximum n nodes.
- Any earlier trees cannot be contained within a newer tree (they are not inf-embeddable within a newer tree). In particular, to check if a tree is inf-embeddable within any later tree, it would (also) need to preserve the nearest common ancestor.
Therefore (going left-to-right, then top-to-bottom), how is tree 3 in the example not in violation of rule 2? It has the "sequence" B-R-B, so the nearest common ancestor is R, and this is clearly contained in trees 5-10 and 12, where I can easily find the subgraph B-R-B with the nearest common ancestor of R. What am I doing wrong?