**Construction of the OSM matrix.** (

**a**) A rooted tree with taxa 1 and 2. (

**b**) A transition

*s*_{1}on the left branch

*e*_{1}(the red branch) changes a character into exactly one new character as depicted by the red horizontal stripe cells of the permutation matrix

. The matrix has 16 rows and 16 columns representing the possible characters for the alignment of two nucleotide sequences. The permutation matrices generated by

*s*_{1}for the right branch

*e*_{2}(blue) and for the branch leading to the “root”

*e*_{12}(green) are displayed in (

**c**) and (

**d**), respectively. The corresponding Cayley graph for the tree is illustrated in (

**e**). The convex sum of all the weighted (by the relative branch length and the probability of the substitution type) permutation matrices generated by all substitution types for all branches is the OSM matrix of the tree (

) as shown in (

**f**). Horizontal stripe cells represent the probability of the transition

*s*_{1}; diagonal stripes the transversion

*s*_{2}; and thin reverse diagonal stripes the transversion

*s*_{3}. The colors of these cells indicate the relative branch lengths and follow the colors of the branches as in (

**a**). Thus, these colors also depict the branch origin of the substitutions.

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