at branching points. The degree of a node is the number of edges incident on that node. We say that a tree is unrooted if it contains no internal nodes with degree less than three. A tree is rooted if it contains a single node with degree two, which we call the root . We do not consider trees with more than one internal node of degree two. The model of character substitution is properly considered a probabilistic graph- ical model defined on the tree. That is, we associate a random variable X$_{s}$ ∈ C with each node s ∈ S so that { X$_{s}$ }$_{s}$$_{∈}$$_{S}$ represents the history of a single column in a multiple sequence alignment. Each X$_{s}$ is independent of all other X$_{r}$ , conditional on the states at the nodes neighbouring s . As we will focus on the assumption of strand symmetry, we will assume that C = { A , C , G , T } . We also assume that each column in the alignment is an independent observation of the multivariate random variable { X$_{s}$ }$_{s}$$_{∈}$$_{T}$ . The model is then specified by a marginal probability row 4-vector π $^{s}$, where π $^{s}$( i ) = P ( X$_{s}$ = i ) for a fixed node s , and a 4 × 4 transition probability matrix P rs for each edge { r, s } ∈ E , where P $^{rs}$( i, j ) = P ( X$_{s}$ = j | X$_{r}$ = i ). As we are interested in rooted tree topologies, it is convenient to characterise the model by π r where r is the root and the P st where { s, t } ∈ E and t is further than s from r . In this context we will sometimes characterise each P st as the result of a continuous-time Markov substitution process on { s, t } , in which case P st = exp Q $^{st}$, where exp is the matrix exponential and Q st is a transition rate matrix. The statistical challenge is then, for a fixed set of terminal nodes T and a set of n observations of { X$_{s}$ }$_{s}$$_{∈}$$_{T}$ , which are the n columns in our multiple sequence alignment, to show that as n tends to infinity our estimates of the tree topology and the probabilistic parameters tend to the true values of the generating model. 7