  For certain HNN extensions including Baumslag-Solitar groups, a treeing is
constructed from their certain probability-measure-preserving actions. This is
a treeing of a quotient groupoid of the translation groupoid associated with
their actions. As its application, for some of those HNN extensions, we show
that the kernel of the modular homomorphism is measure equivalent to the direct
product of the free group of infinite rank and Z.
