A Limiting Theorem for Parsimoniously Bicoloured Trees*

Let T, denote a trivalent tree with n leaves (endnodes) labelled 1,2, . . . , n, and n 2 unlabelled interior nodes of degree three; there are (2n 5)!! = (2n 5)(2n 7) . . .3.1 such trees when n 2 3, a result dating back to 1870 (see [l]). W e suppose that the leaves labelled 1,2,. . . , a are assigned one colour and that the remaining 6 = n a leaves are assigned a second colour. If each interior node of T, is now assigned one of these two colours, then some of the edges of T, will join nodes of different colour (if a, b > 0). The weight w,,I, = wa,b(Tn) of T, is the minimum number of such edges, taken over all the 2n-2 bicolourings of the interior nodes of Tn. Fitch’s algorithm [2] g ives an efficient method for calculating w,,b(T,,). The quantity Wa,b(Tn) is central to the reconstruction of phylogenetic trees from aligned genetic sequences, and for certain applications (for example [3]) ‘t 1 is useful to be able to calculate the probability pa,b(k) that w,,b equals k, taken over all the (2n 5)!! trivalent t,rees T,,. It follows from results of Carter et al. [4] or Steel [5] that: