Adjacency-preserving rooted-tree embedding ($T{\le }_{\mathrm{adj}}S$)

An adjacency-preserving rooted-tree embedding is an injective vertex map preserving the root and every adjacency. Because each root-to-vertex path maps to a simple path of the same length, vertex depths are preserved. This is more restrictive than a homeomorphic embedding of a rooted tree, which may stretch edges into paths. The branching-depth antichain of rooted trees shows that the adjacency relation is not a well-quasi-ordering.

Let $T_m$ consist of a rooted path of $m\ge 1$ edges followed by two leaves at its far end. Its only vertex with two children has depth $m$. An adjacency-preserving rooted-tree embedding must preserve this depth, so $T_m$ embeds into $T_n$ only when $m=n$. This gives an infinite bad sequence, while rooted-tree homeomorphic embeddings can stretch the stems.