On an independence condition for consensus \(n\)-trees (Q922253)

Methods for the aggregation of several tree-structured data into a single ``consensus'' tree are being extensively studied, especially in the field of numerical taxonomy where comparing and constructing trees of various types are everyday activities. A typical example is when several tree construction algorithms, e.g., clustering algorithms, operating on the same data set produce different trees, and it is desired to try to capture the agreement of all these trees by forming a consensus tree. In addition to the actual construction of consensus functions, there have been theorems modeled after Arrow's Impossibility Theorem that show the inconsistency of some reasonably looking axioms. In this note we outline the proof of such a theorem for n-trees (which are a type of hierarchical classification) hereby settling an open problem of \textit{D. A. Neumann} [Math. Biosci. 63, 271--289 (1983; Zbl 0539.62073)].