= Recursively repetition-free labelled trees preserve Dedekind-finiteness
If $D$ is an <infinite Dedekind-finite set>, then $\mathcal T(D)$ is another such <set>. Single-vertex <rooted trees> inject $D$ into it. A countable <sequence> of distinct <rooted trees> gives finite lists of labels by the <depth-first traversal of a tree> in the prescribed child order. By <countable union of explicitly ordered finite lists without choice>, an infinite union of labels contradicts Dedekind-finiteness. A finite union $F$ is equally impossible, since every <rooted tree> then lies in the <finite set> $\mathcal T(F)$, by <mathematical induction> on $|F|$ using the recursive child rule. Thus no countably infinite subset of <rooted trees> exists.
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