[en] This paper evaluates the performance of downlink information transmission in three-dimensional (3D) unmanned aerial vehicle (UAV) networks, where multi-tier UAVs of different types and flying altitudes employ directional antennas for communication with ground user equipments (UEs). We introduce a novel tractable antenna gain model, which is a nonlinear function of the elevation angle and the directivity factor, for directional antenna-based UAV communication. Since the transmission range of a UAV is limited by its antenna gain and the receiving threshold of the UEs, only UAVs located in a finite region in each tier can successfully communicate with the UEs. The communication connectivity, association probability as well as coverage probability of the considered multi-tier UAV networks are derived for both line-of-sight (LoS) and non-line-of-sight (NLoS) propagation scenarios. Our analytical results unveil that, for UAV networks employing directional antennas, a necessary tradeoff between connectivity and coverage probability exists. Consequently, UAVs flying at low altitudes require a large elevation angle in order to successfully serve the ground UEs. Moreover, by employing directional antennas an optimal directivity factor exists for maximizing the coverage probability of the multi-tier UAV networks. Simulation results validate the analytical derivations and suggest the application of high-gain directional antennas to improve downlink transmission in the multi-tier UAV networks.