The structure of a tree¶
Fig. 1 The genealogical relationship of five lineages. It may be simplest to think of a lineage as a haploid lineage. Alternately, each lineage could be one lineage in a diploid. For example, the maternal or the paternal allele. Each node is labelled by birth time, number of descendant nodes, and branch length (int time) to the node’s parent. All branch lengths are in generations. By convention, time zero (0) refers to the present. (Zero generations ago.) Time increases as we move from the present into the past.¶
This tree is the genalogical history of a genomic region of length \(L\) units.
Typically, the units of \(L\) will be in base pairs. But, it could be in more abstract, such as genetic distance.
If we define a mutation rate, \(\mu\) as the expected number of mutations per unit, per generation…
…then the expected number of mutations on a branch is \(B\mu L\), where \(B\) is the branch length.
A mutation on a branch above a node with \(d\) descendants affect the character state of all descendants…
…unless there’s another mutation at the same position on one (or more) descendant lineages.
It is often convenient to measure time in generations divided by the ploidy times the population size.
For example, in a diploid population of \(N\) indidivuals, we may measure time as \(\frac{g}{2N}\).
When we do this, we must also re-scale the mutation rate accordingly. Again for diploids, we’d use \(2N\mu\).