The branch-and-cut paradigm has led to effective solvers for a wide range of mathematical optimization problems. Despite significant theoretical work on pure branching and pure cutting, less is known about their interaction. We study an extended abstract branch-and-cut model to investigate tradeoffs between branching and cutting, including the time to generate cuts. We prove that, under symmetric branching, some minimal-time tree has all cuts at the root, before the first branching node, and that there is an asymmetric setting where some minimal-time tree has no internal cuts, i.e., cuts both preceded and followed by branching. For other settings, although a root-cut tree may be suboptimal and differ substantially in structure from a minimal-time tree, it nevertheless provides a reliable approximation whose tree time does not deviate arbitrarily from the optimum. For constant cut strength and bounded time functions, we show that the best root-cut tree is asymptotically optimal as the target bound grows. In addition, under a single-cut-per-path restriction, we characterize when some minimal-time tree has an internal cut. Lastly, simulations indicate that the best root-cut tree is much closer to a minimal-time tree than our worst-case approximation guarantee suggests.