Capturing a phylogenetic tree when the number of character states varies with the number of leaves

We show that for any two values $\alpha, \beta >0$ for which $\alpha+\beta>1$ then there is a value $N$ so that for all $n \geq N$ the following holds. For any binary phylogenetic tree $T$ on $n$ leaves there is a set of $\lfloor n^\alpha \rfloor$ characters that capture $T$, and for which each character takes at most $\lfloor n^\beta \rfloor$ distinct states. Here `capture' means that $T$ is the unique perfect phylogeny for these characters. Our short proof of this combinatorial result is based on the probabilistic method.Comment: 3 pages, 0 figure

Identifying X-Trees with Few Characters

Previous work has shown the perhaps surprising result that, for any binary phylogenetic tree T, there is a set of four characters that define T. Here we deal with the general case, where T is an arbitrary X-tree. We show that if d is the maximum degree of any vertex in T, then the minimum number of characters that identify T is log2d (up to a small multiplicative constant)

Identifying X-trees with few characters

Previous work has shown the perhaps surprising result that, for any binary phylogenetic tree T, there is a set of four characters that define T. Here we deal with the general case, where T is an arbitrary X-tree. We show that if d is the maximum degree of any vertex in T, then the minimum number of characters that identify T is log₂d (up to a small multiplicative constant)

Identifying X-Trees with Few Characters

Previous work has shown the perhaps surprising result that, for any binary phylogenetic tree T, there is a set of four characters that define T. Here we deal with the general case, where T is an arbitrary X-tree. We show that if d is the maximum degree of any vertex in T, then the minimum number of characters that identify T is log 2 d (up to a small multiplicative constant).