F. R. McMorris

An algorithm to find agreement subtrees

Given two binary trees, a largest subtree contained in both of the original trees that has been obtained by pruning vertices is called an agreement subtree. An exact algorithm for finding an agreement subtree is presented.

The agreement metric for labeled binary trees

Let S be a set of n objects. A binary tree of S is a binary tree whose leaves are labeled without repetition from S. The operation of pruning a tree T is that of removing some leaves from T and suppressing all inner vertices of degree 2 which are formed by this deletion. Given two trees T and U, an agreement tree is a tree that can be obtained from T as well as from U by pruning the fewest number of leaves from the two trees. A quadratic algorithm is presented for doing this and two metrics are defined based on agreement trees.

Axioms for consensus functions on undirected phylogenetic trees

Abstract A version of Arrows impossibility theorem is established for consensus functions on undirected phylogenetic trees.

The computation of consensus patterns in DNA sequences

Two important consensus problems are closely related to two well-known sequence problems. M. Watermans problem of finding consensus strings is a natural extension of the Longest Common Substring problem. The problem of identifying consensus subsequences is a natural extension of the Longest Common Subsequence problem, and thus is NP-hard.

Majority-rule (+) consensus trees

The construction of a consensus tree to summarize the information of a given set of phylogenetic trees is now routinely a part of many studies in systematic biology. One popular method is the majority-rule consensus tree. In this paper we introduce and characterize a new consensus method that refines the majority-rule tree by adding certain compatible clusters satisfying a simple criterion.

Discovering Consensus Molecular Sequences

Consensus methods are valuable tools for data analysis, especially when some sort of data aggregation is desired; but although methods for discovering consensus sequences play a vital role in molecular biology, researchers often seem inattentive to the features and limitations of such methods, and so there are risks that criteria for discovering consensus sequences will be misused or misunderstood. To appreciate better the issues involved, we survey methods for discovering consensus sequences such as those based on frequency thresholds, voting strategies, heuristics, neighbourhoods, and measures of inhomogeneity or information content.

The median procedure for n-trees as a maximum likelihood method

A few axioms are presented which allow the median procedure for n-trees to be given a maximum likelihood interpretation.

Measuring the Phylogenetic Randomness of Biological Data Sets

Two qualitative taxonomic characters are potentially compatible if the states of each can be ordered into a character state tree in such a way that the two resulting character state trees are compatible. The number of potentially compatible pairs (NPCP) of qualitative characters from a data set may be considered to be a measure of its phylogenetic randomness. The value of NPCP depends on the number of evolutionary units (EUs), the number of characters, the number of states in the characters, the distributions of EUs among these states, and the amount and distribution of missing information and so does not directly indicate degree of phylogenetic randomness. Thus, for an observed data set, we used Monte Carlo methods to estimate the probability that a data set chosen equiprobably from among those identical (with respect to all the other above determining features) to the observed data set would have as high (or low) an NPCP as the observed data set. This probability, the realized significance of the observed NPCP, is attractive as an indication of phylogenetic randomness because it does not require the assumptions made by other such methods: No character state trees are assumed and consequently, only potential compatibility can be determined; no particular method of phylogenetic estimation is assumed; and no phylogenetic trees are constructed. We determined the values and significances of NPCP for analyses of 57 data sets taken from 53 published sources. All data sets from 37 of those sources exhibited realized significances of < 0.01, indicating high levels of phylogenetic nonrandomness. From each of the remaining 16 sources, at least one data set was more phylogenetically random. Inclusion of outgroups changed significance in some cases, but not always in the same direction. Data sets with significantly low NPCP may be consistent with an ancient hybrid origin (or other ancient polyphyletic gene exchange, crossing over, viral transfer, etc.) of the study group.

On the consistency of the plurality rule consensus function for molecular sequences

The plurality rule for molecular sequences is a consensus function ℘ which maps each profile of lengthk (i.e., each sequence ofk bases appearing at an aligned position ofk molecules) to a set of consensus results (i.e., ambiguity codes) that are descriptive summaries of the profile. Since the plurality rule ℘ is a median procedure, its results are solutions of an optimization problem that is well known and intensively studied in the theory of social choice. However, ℘s behavior for long profiles according to its behavior for short profiles. Because consistency is a desirable feature of consensus functions, we explore the boundaries of its applicability to ℘. We distinguish between two types of consistency, weak and strong, and we apply them to profiles and consensus results as well as to ℘ itself. For ℘ we obtain simple characterizations of weakly consistent profiles, of weakly consistent results, and of strongly consistent results.

The asymptotic plurality rule for molecular sequences

The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n (i.e., each sequence of n bases appearing at an aligned position of n molecules) to a set apl (P) of consensus results (i.e., ambiguity codes) that is a descriptive summary of P. Our main result is to characterize each consensus result X = apl(P) in terms of the frequencies with which the bases in P occur. We then use these characterizations to investigate features (e.g., strong consistency, length independence) of apl that researchers may find useful for the interpretation of apls consensus results.