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