… a tricky task !? Tanja Gernhard, Klaas Hartmann Simulating trees… … a tricky task !? Tanja Gernhard, Klaas Hartmann              

Even simple models hard to analyze analytically! Why simulations? How do typical trees look like under a specific model of speciation? expected Colless value? expected gamma value, LTT plot? p-values for statistical testing? Even simple models hard to analyze analytically!

Aspects of simulations Tree distribution ill-defined (expected age of the tree is infinite) Condition on age of tree (simulations straightforward) Condition on number of species in the tree (assuming a uniform prior for the age of the tree)

Standard method Simulate until n species are obtained. Stop at following speciation or extinction event. Example for n=5 species: Later periods with n species disregarded! Pendant edges are too long! Each simulation makes same contribution!

General evolutionary model Simulate a tree, t, until n* species or extinction is reached Find the expected number of trees to sample from t: For each sample required: Randomly choose an interval, i, according to the weights Choose the pendant edge length uniformly at random from (0, ) Repeat from step 1 until the required number of samples has been obtained © = P k i 1 Á Á i Á i

Special models Yule model: Pure birth process. Individuals evolve independently. Speciation occurs at a constant rate. Constant rate birth-death model: Add constant death rate to the Yule model. Coalescent: Time between successive coalescent events is exponentially distributed

Memoryless pure birth models Time between sn and sn+1 does not depend on the current tree (e.g. Yule model, Coalescent). Correct simple approach: Simulate until Determine s n : ¸ :

Determining . ¸ h ( ¸ ) / Z 1 j ¾ n g d = Z 1 ¸ g ( ¾ n ) d

Yule model / Coalescent h ( ¸ ) / Z 1 g ¾ n d = f e ¡ Simulating until the (n+1)-th speciation event yields the desired results!

Birth-death model f ( s j t = ; n ) ¸ ¡ ¹ e 1 q ( t j n ) = ¸ ¡ ¹ 1 e reconstructed tree tor point process s3 s1 s2 s4 1 2 3 4 5 f ( s i j t o r = ; n ) ¸ ¡ ¹ 2 e 1 q o r ( t j n ) = ¸ ¡ ¹ 2 1 e +

Available Program Tool TreeSample implemented in Pearl by Klaas Hartmann. Available as a stand-alone application or as a Pearl script. Demonstration:

Dankeschön Dennis Wong Mike Steel Erick Matsen Klaas Hartmann