we give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure preserving equivalence relation. the main philosophy is to understand the behavior of the radon-nikodym cocycle in terms of the geometry of the amenable subrelation within the tree. this allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. the two main ingredients are an extension of carrière and ghys's criterion for nonamenability, along with a new ping-pong-style argument we call the "paddle-ball lemma" that we use to apply this criterion in our setting. this is joint work with anush tserunyan.