This is a follow-up to an earlier post, which showed an example of two phylogenetic trees and three rooted phylogenetic networks. You can see them again in the figure below.
Each of the networks N1, N2 and N3 displays the two trees T1 and T2 (and no other trees). Thus, it is impossible to decide which of the three networks is correct. The question was asked whether this is a fundamental limitation of rooted phylogenetic networks (a.k.a. hybridization networks).
Let's first draw the networks such that each reticulation is an instantaneous event between two coexisting taxa. To do so, networks N2 and N3 need an additional taxon x, which could be an extinct taxon or just a taxon that has not been sampled.
I've specified a length for each edge of each network and have given corresponding edge lengths to the trees. The values of the edge lengths in the networks have been chosen rather arbitrarily, and are not important for the discussion below.
What is important is that, when you take the edge lengths into account, it is easy to decide which of the three networks should be chosen. N1 should be chosen if the roots of T1 and T2 have the same age, N2 should be chosen if the root of T1 is older and N3 if the root of T2 is older. The reason is the following. In network N1, the roots of T1 and T2 both coincide with the root of the network. This contrasts with network N2, where the root of T2 is a proper descendant of the root of T1 and with network N3, in which the root of T1 is a proper descendant of the root of T2.
We can conclude that the above example shows an important challenge but not a fundamental limitation of rooted phylogenetic networks. When taking edge lengths into account, it is indeed possible to uniquely reconstruct the network (at least in this case).