Evolutionary histories are often described by phylogenetic trees. However, reticulate events such as hybridisation and horizontal gene transfer cannot be described by phylogenetic trees alone. Histories that include these events require phylogenetic networks. For reconstructing evolutionary histories, an important question is whether a network or a tree is needed to describe such a history. Given sufficiently long DNA sequences, it has previously been established that level-1 and level-2 three-leaf networks can be distinguished from threeleaf trees using a phylogenetic invariant. Here, we generalise this result to networks of level-k, where k ≥ 1, and show the same distinguishability holds for networks on more than three leaves. This makes it, at least in theory, possible to determine whether the evolutionary history of different taxa happened in a network-like or tree-like way.