** Next:** Probabilistic Dynamic Programming and
** Up:** Probability of an evolutionary
** Previous:** How to compute this

It can be shown that the probability of an EC
is independent of where we place the root of the tree.
This is a direct consequence of the symmetry of the evolution model
mentioned in 1.1.
To prove this, we first show that the probabilities are not
affected if we move the root along the edge liking its two
descendants (*X* and *Y*).
The length of the branch from *X* to *Y* will remain constant,
we will just change *d*_{X} and *d*_{Y} so that
.
The probability of the EC is

Moving the summations around and using the symmetry relation
of *f* and *M* and noticing that the summation in *i* gives
a matrix product, we obtain

and the formula is independent of *d*_{X} and *d*_{Y}, only depends on
their sum.
By continuity, this can be extended to any position within the EPT.
Consequently for the purposes of evaluating the probability of
a particular EC, we can place the root
wherever it is most convenient.

** Next:** Probabilistic Dynamic Programming and
** Up:** Probability of an evolutionary
** Previous:** How to compute this
*Gaston Gonnet*

*1998-07-14*