A note on convex characters and Fibonacci numbers

Abstract

Given an unrooted, binary phylogenetic tree T on a set of n >= 2 taxa, a closed expression for the number of convex characters on T has been known since 1992, and this is independent of the exact topology of T. Here we prove that this number is equal to the (2n-1)th Fibonacci number. Moreover, we show that the number of convex characters in which each state appears on at least two taxa, is also independent of topology, and equal to the (n-1)th Fibonacci number. We use this insight to give a simple but effective algorithm for the NP-hard "maximum parsimony distance" problem that runs in time \(\Theta( \phi^{n} \cdot \text{poly}(n) )\), where \(\phi \approx 1.618...\) is the golden ratio. Finally, we give an explicit example demonstrating that topological neutrality no longer holds when counting the number of convex characters in which each state appears on at least three taxa.