Non-explosion principles for branched rough differential equations with unbounded coefficients

Expanding upon the work of arXiv:2502.08799, we provide a non-explosion criterion for branched rough differential equations (bRDEs) where we continue to allow for the drift and the rough coefficient to be unbounded in the branched setting. Moreover, by providing a characterization of ``pure area'' branched rough paths -- branched rough paths over the zero path, we provide two different constructions of bRDEs which explode in finite time, and thus demonstrating the sharpness of the criterion. Finally, by realizing a trade-off between the growth of the coefficient of the bRDE and the decay of its higher-order derivatives, we provide a new non-explosion principle for bRDEs which allows for the coefficient to grow even faster than what is provided in arXiv:2502.08799.