A Complete Characterization of Pairs of Binary Phylogenetic Trees with Identical $$A_k$$-Alignments

Abstract Phylogenetic trees play a key role in the reconstruction of evolutionary relationships. Typically, they are derived from aligned sequence data (like DNA, RNA, or proteins) using optimization criteria like, e.g., maximum parsimony (MP). It is believed that the latter is able to reconstruct the “true” tree, i.e., the tree that generated the data, whenever the number of substitutions required to explain the data with that tree is relatively small compared to the size of the tree (measured in the number n of leaves of the tree, which represent the species under investigation). However, reconstructing the “correct” tree i.e., the tree that generated the data, from any alignment first and foremost requires the given alignment to perform differently on said tree than on others. A special type of alignments, namely so-called $$A_k$$ A k -alignments, has gained interest in recent literature. These alignments consist of all binary characters (“sites”) which require precisely k substitutions on a given tree. It has been found that whenever k is small enough (in comparison to n ), $$A_k$$ A k -alignments uniquely characterize the trees that generated them. However, recent literature has left a significant gap between $$n\\leqslant 2k+2$$ n ⩽ 2 k + 2 – namely the cases in which no such characterization is possible – and $$n\\geqslant 4k$$ n ⩾ 4 k – namely the cases in which this characterization works. It is the main aim of the present manuscript to close this gap, i.e., to present a full characterization of all pairs of trees that share the same $$A_k$$ A k -alignment. In particular, we show that indeed every binary phylogenetic tree with n leaves is uniquely defined by its $$A_k$$ A k -alignments if $$n\\geqslant 2k+3$$ n ⩾ 2 k + 3 . By closing said gap, we also ensure that our result is optimal. Moreover, we show that two trees T and $$T'$$ T ′ have the same $$A_k$$ A k -alignment if and only if $$T'$$ T ′ can be obtained from T using so-called NNI (nearest neighbor interchange – a well-known tree rearrangement operation) moves that additionally obey certain parity constraints. In other words, trees with identical $$A_k$$ A k -alignments are connected in the NNI tree space by paths with specific properties.