running head: COUNTING DUPLICATION TREES The Combinatorics of Tandem Duplication Trees

We develop a recurrence relation that counts the number of Tandem Duplication Trees (either rooted or unrooted) that are consistent with a set of n tandemly repeated sequences generated under the standard unequal recombination (or crossover) model of tandem duplications. We find that the number of rooted duplication trees is exactly twice the number of unrooted trees, which means, on average, only two positions for a root on a duplication tree are possible. Using the recurrence we can tabulate these numbers for small values of n. Further we develop an asymptotic formula, that for large n, provides estimates for these numbers. These numbers give a priori probabilities for phylogenies of the repeated sequences to be duplication trees. This extends earlier studies where exhaustive counts of the numbers for small n were obtained. One application showed the significance of finding that most maximum parsimony trees constructed from repeat sequences from Human immunoglobins and T-cell receptors were tandem duplication trees. Those findings provided strong support to the proposed mechanisms of tandem gene duplication. The recurrence relation also suggests efficient algorithms to recognize duplication trees and to generate random duplication trees for simulation. A linear-time recognition algorithm is detailed. [