Using a nonrecursive formula to determine cladogram probabilities

Three properties of bifurcating branching diagrams that are used for representing a specific number of taxa are (1) the number of possible arrangements, (2) the number of possible topologies, and (3) the probabilities of formation according to particular models of cladogenesis. Of these, the probabilities have received the least attention in the literature. Indeed, many biologists would be astonished by the observation that the probability of a commonly cited cladogram containing 35 phyla of the animal kingdom is < 0.0072% of the value of the average probability taken over all possible cladograms! We reviewed works on cladogram arrangements and topologies and developed a computer-generated table of enumerations that extends and corrects such tables in the literature. We also developed a nonrecursive formula for the determination of cladogram probabilities. This formula facilitates calculation and thereby should promote use of cladogram probabilities, which might provide more accurate null hypotheses for tests of cladogenic events than do considerations of cladogram arrangements or topologies.