 Solution

The HNN extension is the Baumslag-Solitar groupIts Bass-Serre tree has vertices $G/⟨u⟩$ and oriented edges $G/⟨u⟩$, with the two endpoint maps induced by the identity embedding and the index-two embedding $n\mapsto 2n$. 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.