Small Phylogenetic Trees -- ASCB Conventions

The labellings

If T is a rooted tree, there is a distinguished vertex of T called the root and labeled by the letter r. The tree T should be drawn with the root r at the top of the figure and the edges of the tree below the root. Each edge in the tree is labeled with a lowercase letter a,b,c, ... The edges are labeled in alphabetical order starting at the upper left hand corner, proceeding left to right and top to bottom. The leaves are labeled with the numbers 1,2,3, ... starting with the left-most leaf and proceeding left to right.

If T is an unrooted tree, it should be drawn with the leaves on a circle. The edges of T are labeled with lower-case letters a,b,c, ... in alphabetical order starting at the upper left hand corner of the figure and proceeding left to right and top to bottom. The leaves are labeled with the numbers 1,2,3,... starting at the first leaf "left of 12 o'clock" and proceeding counterclockwise around the perimeter of the tree.

The random variables

For each node in a model, we associate a random variable with two or four states depending on whether we are looking at binary data or DNA data. In the case of binary data these states are {0,1} and for DNA data they are {A,C,G,T} in this order.

The root distribution

The root distribution is a vector of length two or four depending on whether the model is for binary or DNA sequences. The name of this vector is r. Its entries are parameters r₀, r₁, r₂, ... and are filled in from left to right and are recycled as the model requires.

The transition matrices

In each type of model, the letters a,b,c, ... that label the edges represent the transition matrices in the model. These are either 2×2 or 4×4 matrices depending on whether the model is for binary data or DNA data. In each case, the matrix is filled from left to right and top to bottom with unknown parameters, recycling a parameter whenever the model requires it. For the transition matrix of the edge labeled with x, these entries are called x₀, x₁, x₂, ... We assume that the entries of these matrices satisfy additional linear constraints such that they are transition matrices. We do not, however, use these linear relations to eliminate parameters.