We have examined the newly introduced Underhill-Doyle spring to investigate the problem of the motion of star polymers and the first four generations of dendrimers in a viscous liquid (solvent) under the influence of extensional flow. The famous Rouse (bead-spring) model is considered for all branches and Brownian dynamics simulation technique is implemented for simulations. Results on star polymers in transient extensional flow reveal that the initial transient stress response at small strains is governed by both number of arms and the size of the shortest arm. Furthermore, the steady state behavior of star polymers in extensional flow is limited by the maximum effective contour length of the molecules and the effect of excluded volume interactions on the dynamics of star polymers appears unimportant. Under extensional flow, branched polymers present the typical coil-stretch transition when the extensional rate exceeds a critical value, analogous to the behavior presented by linear chains. We have determined the power law that relates the critical extensional rate to characteristic molecular weight of star polymers and dendrimers. They differ in that critical strain rate of star polymers scales with the arm molecular weight, whereas for dendrimers it scales with the total molecular weight. The critical Wissenberg number adopts a similar value for both topologies around 0.5, and this value seems to be independent of the topology. Also, we present an analysis of the distribution of coil-stretch transition under extensional flow for different polymers and show for extensional rates much larger than the critical values, all the results overlape and become independent of topology.