A Fourier Inversion Formula for Evolutionary Trees

We establish a pair of identities, which will provide a useful tool in the reconstruction of evolutionary trees in Kimura's 3-parameter model. The starting point of this paper was an attempt for a better understanding and generalization of an Hadamard inverse pair of formulae, which was used in statistics by Cooper [l], in image processing by Andrews [2, Chapters 6,7], and in information theory by Whelchel and Guinn [3]. Recently, Hendy and Penny [4] applied this technique to the spectral analysis of phylogenetic data, for two-state character sequences, and they asked for generalization to four-state character sequences, which is the form of a nucleotide sequence. The most invaluable tool for our work was [5], where discrete Fourier analysis is applied to somewhat similar problems. Let us start with the original theorem of Hendy and Penny [6]. Suppose we are given a tree T with leaf set L, out of which one is a root called R. Set ILI = n. Toss a coin independently for every edge e of the tree with probability pe for a head and probability 1-pe for a tail. Let u denote a subset of L \ {R}. Let fc denote the probability that u is precisely the set of leaves for which the unique RI path contains an odd number of edges with head outcome. Let X be a subset of L of even cardinality. Match the vertices of X somehow with paths in the tree and notice that the set of edges P(T, X) used by an odd number of paths is independent of the matching. Set rx= J-J (l-2p,) eEP(T,X) Then, from [S] one has the following formulae: TX = c (-1)'aox' fo, oCL\tW f. = & c (-lp " Xbx.