KöMaL 2026 A934

Let $\mathcal{T}$ denote the set of countable, undirected trees. A subset $X$ of the real numbers is called bariton if every nonempty subset of $X$ has a least element. a) Prove that for every bariton $X$ there exists a function $f\colon X\to \mathcal{T}$ such that for all $y$, $z\in X$ we have $y\leq z$ if and only if $f(y)$ is isomorphic to a subgraph of $f(z)$. b) Does the statement remain true if we additionally require that for every $x\in X$, all vertices of $f(x)$ have finite degree? (A graph with vertex set $V$ is called countable if there exists an injective map $V\to\mathbb{N}$.)