Phylogenetic trees and networks are both used in the scientific literature to display DNA sequence variation at the intraspecific level. Should we rather use trees or networks? I argue that the process of inferring the most parsimonious genealogical relationships among a set of DNA sequences should be dissociated from the problem of displaying this information in a graph. A network graph is probably more appropriate than a strict consensus tree if many alternative, equally most parsimonious, genealogies are to be included. Within the maximum parsimony framework, current phylogenetic inference and network-building algorithms are both unable to guarantee the finding of all most parsimonious (MP) connections. In fact, each approach can find MP connections that the other does not. Although it should be possible to improve at least the maximum parsimony approach, current implementations of these algorithms are such that it is advisable to use both approaches to increase the probability of finding all possible MP connections among a set of DNA sequences.