In this paper, we examine the melt rheology of well-defined, model polymers where the long chain branching (LCB) is precisely known from the synthesis. All of these are made by the hydrogenation of polybutadiene, but they vary greatly in the level and type of LCB present. We find that ail polymers that have LCB show a greater degree of shear thinning than linear chains. This applies both to those with a single branch (stars) and also to those with multiple branches per chain (such as combs). However, only molecules with multiple branches induce extensional thickening in a sample. Only a small amount of these comblike molecules, on the order of 5%, are needed to show this effect. We also show here how a new method of treating the shear data, the so-called Van Gurp-Palmen analysis, can give a more easily interpreted form of the results that can reveal the length and amount of branches in a sample. The insights generated from this work show the importance of access to well-defined polymers with several kinds of branching architecture for the development of a deeper understanding of polymer rheology.