1. A statistic summarizing: Relative timing in trees Tanja Gernhard joint work with Erick Matsen, Daniel Ford

2. Tree statistics Investigate the properties of reconstructed trees Colless [Colless, 1982] summarizes the information of the shape of the tree (i.e. balanced vs. unbalanced) Gamma [Pybus, Harvey 2000] summarizes the time between speciation events (graphical representation in LTT plots)

3. Relative Timing Neutrality: Each ranking equally likely

4. How to summarize rank information? Each ranking equally likely equivalent to For each vertex: Each shuffle equally likely shuffles on a vertex RRRRLLRRLRRL

5. Runs summarize shuffle Runs: Number of sequences in shuffle with same letter Consider (2,4)-shuffle RRLRRL: 4 runs Run distribuition of (m,n)-shuffle well- known from statistics: P f X m ; n = 2 k + 1 g = ¡ m ¡ 1 k ¢¡ n ¡ 1 k ¡ 1 ¢ + ¡ m ¡ 1 k ¡ 1 ¢¡ n ¡ 1 k ¢ ¡ m + n m ¢ P f X m ; n = 2 k g = 2 ¡ m ¡ 1 k ¡ 1 ¢¡ n ¡ 1 k ¡ 1 ¢ ¡ m + n m ¢

6. Combining all shuffles in tree Runs of a tree, R(T): Sum of runs of all shuffles in tree Distribution for runs in tree T: P f R ( T ) = k g = k X i = 1 P f X m ; n = i g k ¡ i X j = 0 P f R ( L ) = j g P f R ( R ) = k ¡ i ¡ j g

7. Application to Ant lineages We investigated the tree of Ants [Moreau et al, 2007] LTT plot show substantial increase during Late Cretacceous to Early Eocene

9. Some comments Testing the posterior of Bayesian methods for neutrality (phylogenies or populations) Population size might change arbitrary All programs available from my homepage in the Python package Cass python Stats.dat YourTrees.tre

10. Dankeschön