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A consensus tree is a phylogenetic tree that summarizes the evolutionary relationships inferred from a collection of phylogenetic trees with the same set of leaf labels. Among the many types of consensus trees that have been proposed in the last 50 years, the frequency difference consensus tree is one of the more finely resolved types that retains a large amount of information. This paper presents a new deterministic algorithm for constructing the frequency difference consensus tree. Given k phylogenetic trees with identical sets of n leaf labels, it runs in O(knlog{n}) time, improving the best previously known solution.

A multi-labelled tree (or MUL-tree) is a rooted tree leaf-labelled by a set of labels, where each label may appear more than once in the tree. We consider the MUL-tree Set Pruning for Consistency problem (MULSETPC), which takes as input a set of MUL-trees and asks whether there exists a perfect pruning of each MUL-tree that results in a consistent set of single-labelled trees. MULSETPC was proven to be NP-complete by Gascon et al. when the MUL-trees are binary, each leaf label is used at most three times, and the number of MUL-trees is unbounded. To determine the computational complexity of the problem when the number of MUL-trees is constant was left as an open problem. Here, we resolve this question by proving a much stronger result, namely that MULSETPC is NP-complete even when there are only two MUL-trees, every leaf label is used at most twice, and every MUL-tree is either binary or has constant height. Furthermore, we introduce an extension of MULSETPC that we call MULSETPComp, which replaces the notion of consistency with compatibility, and prove that MULSETPComp is NP-complete even when there are only two MUL-trees, every leaf label is used at most thrice, and every MUL-tree has constant height. Finally, we present a polynomial-time algorithm for instances of MULSETPC with a constant number of binary MUL-trees, in the special case where every leaf label occurs exactly once in at least one MUL-tree.

A consensus tree is a phylogenetic tree that captures the similarity between a set of conflicting phylogenetic trees. The problem of computing a consensus tree is a major step in phylogenetic tree reconstruction. It is also central for predicting a species tree from a set of gene trees, as indicated recently in [Nature 2013]. This paper focuses on two of the most well-known and widely used consensus tree methods: the greedy consensus tree and the frequency difference consensus tree. Given k conflicting trees each with n leaves, the previous fastest algorithms for these problems were O(k n^2) for the greedy consensus tree [J. ACM 2016] and O~(min{k n^2, k^2n}) for the frequency difference consensus tree [ACM TCBB 2016]. We improve these running times to O~(k n^{1.5}) and O~(k n) respectively.

The problem of constructing a minimally resolved phylogenetic supertree (i.e., having the smallest possible number of internal nodes) that contains all of the rooted triplets from a consistent set R is known to be NP-hard. In this paper, we prove that constructing a phylogenetic tree consistent with R that contains the minimum number of additional rooted triplets is also NP-hard, and develop exact, exponential-time algorithms for both problems. The new algorithms are applied to construct two variants of the local consensus tree; for any set S of phylogenetic trees over some leaf label set L, this gives a minimal phylogenetic tree over L that contains every rooted triplet present in all trees in S, where ``minimal'' means either having the smallest possible number of internal nodes or the smallest possible number of rooted triplets. The second variant generalizes the RV-II tree, introduced by Kannan, Warnow, and Yooseph in 1998.

This paper presents a fast algorithm for finding the Adams consensus tree of a set of conflicting phylogenetic trees with identical leaf labels, for the first time improving the time complexity of a widely used algorithm invented by Adams in 1972 [1]. Our algorithm applies the centroid path decomposition technique [9] in a new way to traverse the input trees' centroid paths in unison, and runs in O(k n \log n) time, where k is the number of input trees and n is the size of the leaf label set. (In comparison, the old algorithm from 1972 has a worst-case running time of O(k n^2).) For the special case of k = 2, an even faster algorithm running in O(n \cdot \frac{\log n}{\log\log n}) time is provided, which relies on an extension of the wavelet tree-based technique by Bose et al. [6] for orthogonal range counting on a grid. Our extended wavelet tree data structure also supports truncated range maximum queries efficiently and may be of independent interest to algorithm designers.

Consider a set of labels $L$ and a set of trees ${mathcal T} = { {mathcal T}^{(1), {mathcal T}^{(2), ldots, {mathcal T}^{(k) $ where each tree ${mathcal T}^{(i)$ is distinctly leaf-labeled by some subset of $L$. One fundamental problem is to find the biggest tree (denoted as supertree) to represent $mathcal T}$ which minimizes the disagreements with the trees in ${mathcal T}$ under certain criteria. This problem finds applications in phylogenetics, database, and data mining. In this paper, we focus on two particular supertree problems, namely, the maximum agreement supertree problem (MASP) and the maximum compatible supertree problem (MCSP). These two problems are known to be NP-hard for $k geq 3$. This paper gives the first polynomial time algorithms for both MASP and MCSP when both $k$ and the maximum degree $D$ of the trees are constant.