Branching-depth antichain of rooted trees
= Branching-depth antichain of rooted trees
{title2=$T_1,T_2,\ldots$}
Let $T_m$ consist of a rooted path of $m\geq1$ 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.