  Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$
treeable. We show that for any treeable action of $G/C$ on a standard
probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct
product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply
this to show that any group in the minimal class of groups containing treeable
groups and closed under taking direct products, commensurable groups and
central extensions has the Haagerup property.
