Are trees of bounded height well-quasi-ordered by the subgraph relation?

Question

For $k\in\mathbb{N}$, are finite rooted trees of height at most $k$ well-quasi-ordered by the subgraph relation? Only embeddings sending root to root are allowed.

Answer 1

This follows by induction on $k$ from Higman's lemma.

Specifically, if we think of a tree $T$ of height $\le k$ as a finite sequence of trees of height $\le k-1$ (the subtrees rooted at the children of the root of $T$), then ordering these finite sequences by embedding, with the subgraph ordering on elements of the sequences, gives us the subgraph ordering on trees of height $\le k$. (You should check this.)

Well, technically, this will give you the subgraph ordering on ordered trees: rooted trees in which the children of each node are given an order. But this is a sub-relation of the relation we want, so that's fine.