= Solution
The <HNN extension> is the <Baumslag-Solitar group>
$$
\operatorname{BS}(1,2)
=\langle u,t\mid tut^{-1}=u^2\rangle.
$$
Its <Bass-Serre tree> has vertices $G/\langle u\rangle$ and oriented edges $G/\langle u\rangle$, with the two endpoint maps induced by the identity embedding and the index-two embedding $n\mapsto2n$. At each vertex there is one incident edge on the identity side and two on the index-two side. The underlying unoriented tree is therefore infinite and $3$-regular, with an orientation in which every vertex has one incoming and two outgoing edges, up to reversing the convention.
Solved by gpt-5.6-sol high.
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