  This work is concerned with forest and cumulant type expansions of general
random variables on a filtered probability spaces. We establish a "broken
exponential martingale" expansion that generalizes and unifies the
exponentiation result of Al{\`o}s, Gatheral, and Radoi\v{c}i\'c and the
cumulant recursion formula of Lacoin, Rhodes, and Vargas. Specifically, we
exhibit the two previous results as lower dimensional projections of the same
generalized forest expansion, subsequently related by forest reordering. Our
approach also leads to sharp integrability conditions for validity of the
cumulant formula, as required by many of our examples, including iterated
stochastic integrals, L\'evy area, Bessel processes, KPZ with smooth noise,
Wiener-It\^o chaos and "rough" stochastic (forward) variance models.
