Defining a phylogenetic tree with the minimum number of small-state characters

Abstract

Phylogenetic trees represent evolutionary relationships and can be uniquely defined by sets of finite-state biological characteristics. Despite prior work showing that sufficiently large trees can be determined by $r$-state character sets, the minimal leaf thresholds $n_r$ remain largely unknown. In this work, we establish the 3-state case as $n_3 = 8$, providing a concrete base for higher-state analyses. We then resolve the 5-state problem by constructing a counterexample for $n=15$ and proving that for $n \geq 16$, $\lceil (n-3)/4 \rceil$ 5-state characters suffice to uniquely define any tree. Our approach relies on rigorous mathematical induction with complete verification of base cases and logically consistent inductive steps, offering new insights into the minimal conditions for character-based tree identification.