Extensive Brownian dynamics simulations of conformational changes accompanying the overcharging of a dendrimer by an oppositely charged long linear polyelectrolyte (LPE) have been carried out. The simulated results have been compared with the predictions of the Nguen and Shklovskii correlation theory [Physica A 293, 324 (2001)] for impenetrable charged spherical macroion. Dendrimer overcharging is caused by the spatial correlations between the "excess" of the LPE charges adsorbed onto its surface. The simulated LPE-length dependence of the corresponding "correlation" energy is in agreement with the theoretical predictions. Maximum of the LPE adsorption occurs at some critical LPE length N{ch};{c} , and the first order phase transition from completely coiled conformation to the conformation with released tails takes place. The phase transition is accompanied by the drastic increase in the relative fluctuations of the polyelectrolyte size. Upon increasing the linear-chain length above N{ch};{c} , the one-long-tail conformation becomes energetically preferable; the exchange time between the long-tail conformation and the short-tail conformation is very large.