Phylogenetic inference is a fundamental aspect of evolutionary biology, with the recoverabilityof the species tree being crucial to the statistical consistency of algorithms that utilize certain statistical quantities. Additionally, it is important to analyze the theoretical sample complexity of statistically consistent algorithms. In this thesis, we summarize our results in three different directions regarding recoverability and sample complexity: 1. For a widely used species tree to gene tree model, can we extend its recoverability result to a more general phylogenetic structure? 2. For a statistically consistent species tree inference algorithm, can we find a good theoretical upper bound for sample complexity? 3. Can we use similar tools for providing recoverability results from a well-known model to other models? For the first question, we consider phylogenetic networks, which is the generalized notion of phylogenetic trees to model non-vertical inheritance, by which a lineage inherits genetic material from multiple parents through hybridization or other lateral gene transfer. Under the network multispecies coalescent (NMSC) model, any topology can be observed in individual gene trees arising from a network. In Chapter 3, we provide a recoverability result for level-1 networks under NMSC through expected internode distances. This result can be considered as an extension of the recoverability of species trees under the multispecies coalescent (MSC) model. The global and local features of our identifiability result are consistent with other studies about the level-1 network with or without NMSC (i.e. [1] and NANUQ in [2]). For the second question, we make progress towards deriving sample complexity bounds for our new species tree reconstruction methods based on sample averages of internode distances. Our sample complexity depends polylogarithmically on N, the number of species in the tree. Our results are similar to sample complexity results for other stochastically consistent species tree inference algorithms, i.e., GLASS (Mossel and Roch, [3]) and ASTRAL (Mirarab et al. [4] and Shekhar et al. [5]), although do not quite match them. For the last question, we consider the gene duplication and loss (GDL) model and estimate the species trees from multilocus data sets. As a top-down model, GDL explains the phenomenon of gene tree incongruence by gene duplication and loss. In Chapter 5, we derive a theoretical result using expected internode distance to recover the unrooted species tree topology under the GDL model in the loss-less case. As far as we know, this is the first theoretical result about the recoverability of the species tree using distance-based methods under the GDL model.