LETTER On Counting Tandem Duplication Trees

Large genomes are full of repeated DNA sequences. It was estimated that over half of the human DNA consists of repeated sequences (Baltimore 2001; Eichler 2001; Leem et al. 2002). Tandem duplication is one of the important evolutionary mechanisms for producing repeated DNA sequences, in which the copies that may or may not contain genes are adjacent along the genome. Fitch (1977) first observed that tandem duplication histories are much more constrained than speciation histories and proposed to model them assuming that unequal crossover is the biological mechanism from which they originate. The corresponding trees are now called tandem duplication trees, the term tandem sometimes being omitted for the sake of conciseness. With more and more genomic sequences becoming known, inferring tandem duplication history has again redrawn researchers’ attention (Benson and Dong 1999; Tang, Waterman, and Yooseph 2002; Elemento, Gascuel, and Lefranc 2002; Zhang et al. 2003). The aim of this article is, first, to present a simple recurrence formula for the number of rooted duplication trees based on the recurrence formula proved in Gascuel et al. (2003). We also give a simple non-counting proof of the fact that the number of rooted duplication trees for n segments is exactly twice the number of unrooted duplication trees for n segments. Notice that this fact was proved based on a counting argument in Gascuel et al. (2003).