Phylogenetic networks can provide a more complete description of evolutionary history than trees, by allowing reticulate events such as hybridization and lateral gene transfer. In the first part of this talk, we explore the question: ‘when is a phylogenetic network merely a tree with additional links between its edges?’ It turns out that the class of ‘tree-based’ networks can be efficiently characterized. More recent results on this question have followed, motivated by Dilworth’s theorem (for posets), and matching theory in bipartite graphs. This allows for fast algorithms to determine when a network is tree-based and, if not, to calculate how ‘close’ to tree-based it is. In the second part of the talk, we model lateral gene transfer by a simple stochastic process on trees. By connecting this process to a simple random walk on a graph it is possible to analyze the extent to which an underlying species tree T can be inferred from sampled gene trees that have undergone random transfers on T.