We can additionally require that they appear as new vertices in terms with isomorphic new parts. Even the barest form of this conclusion is enough to require large cardinals. I don't know how to get independence results for construction sequences of trees, where a single new vertex is introduced at each stage, as in the previous two sections. However, I do have confidence that I can do this when I allow multiple entries of the same vertex. A tree is a partial ordering with a minimum element, where the set of predecessors of every vertex is linearly ordered.
Analogous results hold starting from a given tree.
We can furthermore require that they also appear as new labels in terms with isomorphic new parts. Here isomorphisms between trees are order preserving bijections between the underlying integers which preserve the relevant structure. We can furthermore require that they also appear as new labels in terms with isomorphic new parts. This should be equivalent to the 1-consistency of type theory, or Zermelo set theory with bounded separation. But this requires a yet more profound reworking of the Annals paper.