The spreadsheet gives the probabilities of the 15 possible gene trees for 4 taxa when the species tree has the form (1,(2,(3,4))) - for those who aren't familiar with this notation, this is the asymmetric four-taxon tree with tips 3 and 4 most closely related, then taxon 2, and then taxon 1. There are two internal branch lengths in this tree. The branch connecting the clade containing 3 and 4 to taxon 2 is denoted by y in the spreadsheet, and the internal branch coming directly off the root of the tree is denoted by z.

When you open the spreadsheet, you should enable macros. This will allow you to run a macro which shows the effect of decreasing the length of branch z, while holding the length of branch y constant at 1.0 (in coalescent units). In the top right corner of the chart, you can see the value of branch length z. The bars in the chart correspond to the 15 trees, and the heights of the bars give the probability of that gene tree occurring under the coalescent model with the specified branch lengths.

To run the macro, hit ctrl-p, and watch the magic! Ok, it's really not too exciting, but you should keep your eye on what happens to the first bar as the length of z decreases. Notice that this first bar is the probability of the gene tree that matches the species tree. The probability decreases as z decreases, until finally, when the animation stops at z=0.001, this tree is no longer the most probable.

A good exercise to test your understanding is to see if you can figure out why this tree isn't the most probable. Start with "When z is short, it is unlikely for coalescence to occur on this branch, and so the lineages available to coalesce prior to the root are .... and the coalescent model says ...." (we'll go over it in class this Friday, too).

In addition to just playing with it and having fun, there are two other scenarios to try that are informative. First, try making both y and z small, and see what you get (to do this, move the bar chart over, and just enter new values in the cells for y and z, b3 and c3. The chart will automatically update). See if you can justify why it makes sense - what does the tree look like when y and z are both small? Then try making both y and z large (> 2.0) - this is the more common situation for most taxa, and what you see on the bar chart should make the biologist in you happy.

As always, let me know if there are questions. Enjoy!