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We consider exact enumerations and probabilistic properties of ranked treeswhen generated under the random coalescent process. Using a new approach, basedon generating functions, we derive several statistics such as the exactprobability of finding k cherries in a ranked tree of fixed size n. We thenextend our method to consider also the number of pitchforks. We find arecursive formula to calculate the joint and conditional probabilities ofcherries and pitch- forks when the size of the tree is fixed.

Exact enumeration of cherries and pitchforks in ranked trees under the coalescent model