This paper establishes alternative characterizations of very expressive classes of existential rule sets with decidable query entailment. We consider the notable class of greedy bounded-treewidth sets (gbts ) and a new, generalized variant, called weakly gbts (wgbts ). Revisiting and building on the notion of derivation graphs, we define (weakly) cycle-free derivation graph sets ((w) cdgs ) and employ elaborate proof-theoretic arguments to obtain that gbts and cdgs coincide, as do wgbts and wcdgs. These novel characterizations advance our analytic proof-theoretic understanding of existential rules and will likely be instrumental in practice.