Two-tier femtocell networks have been shown to provide superior performance in terms of indoor user coverage and data rate as compared to conventional macrocellular networks. Nevertheless, when macrocells and femtocells share the whole spectrum, macrocellular communication links suffer severely from inter-tier interference, further exacerbated in dense femtocell network deployments. Therefore, it is of interest to investigate optimal subchannel allocation strategies and characterize their dependence on network density in two-tier femtocell networks. In this paper, we consider a joint subchannel scheme, in which the whole spectrum is shared by both tiers, as well as a disjoint subchannel scheme, whereby disjoint sets of subchannels are assigned to each tier. We derive analytical expressions for the performance of different spectrum allocation schemes in terms of success probability and per-tier network throughput. Furthermore, we formulate the aggregate network throughput maximization problem subject to quality of service constraints in terms of success probabilities and per-tier minimum rate, and provide practically relevant solutions and insights on the optimal spectrum allocation scheme. Our results indicate that with closed access femtocells, the optimized joint and disjoint subchannel schemes provide the highest throughput among all schemes in sparse and dense femtocell network, respectively.