it contains all matrices M such that for every permutation matrix R = I , we have RM / ∈ M . Note that if M is reconstructible from rows then M ⊆ S ( M ). To slightly abuse the terminology, sympathetic matrices are useful because they provide matrices that might not be reconstructible from rows, but that will not contradict the ordering of states of internal nodes that is suggested by matrices that are. Example 3.4. The second matrix in Example 3.3 is not DLC but is from the set of matrices that is sympathetic to the set of DLC matrices. The following theorem extends and restricts Theorem 4.1 in Chang (1996). It makes an additional assumption of strand symmetry, but in so doing is able to accommodate rooted topologies. Motivated by Remark 4 in Chang (1996), it relaxes the assumption that every edge must be reconstructible from rows. Theorem 4.1 in Chang (1996) allows degenerate topologies, where nodes are allowed to have degree greater than three. We assume nondegenerate topologies for ease of exposition and because we will only make that assumption later anyway when we prove that the model can be statistically consistently estimated, as Chang does in his Theorem 5.1. We note that extension of the following theorem to degenerate topologies should be straightforward. Theorem 3.1. Assume the conditions for Corollary 3.1. Define M as a set of matrices that is reconstructible from rows and S ( M ) as the set of matrices that are sympathetic to M . Further assume that (E) there exists a path from every internal node m to a terminal node such that for each edge { u,v } in the path, where u is closer to m than v , P uv ∈ M , and that (F) P uv ∈ S ( M ) for every edge { u,v } in the topology. 16