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The Minimal Ancestral Deviation (MAD) method is a recently introduced procedure for estimating the root of a phylogenetic tree, based only on the shape and branch lengths of the tree. The method is loosely derived from the midpoint rooting method, but, unlike its predecessor, makes use of all pairs of OTUs when positioning the root. In this note we establish properties of this method and then describe a fast and memory efficient algorithm. As a proof of principle, we use our algorithm to determine the MAD roots for simulated phylogenies with up to 100,000 OTUs. The calculations take a few minutes on a standard laptop.
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Given a gene tree and a species tree, ancestral configurations represent the combinatorially distinct sets of gene lineages that can reach a given node of the species tree. They have been introduced as a data structure for use in the recursive computation of the conditional probability under the multispecies coalescent model of a gene tree topology given a species tree, the cost of this computation being affected by the number of ancestral configurations of the gene tree in the species tree. For matching gene trees and species trees, we obtain enumerative results on ancestral configurations. We study ancestral configurations in balanced and unbalanced families of trees determined by a given seed tree, showing that for seed trees with more than one taxon, the number of ancestral configurations increases for both families exponentially in the number of taxa $n$. For fixed $n$, the maximal number of ancestral configurations tabulated at the species tree root node and the largest number of labeled histories possible for a labeled topology occur for trees with precisely the same unlabeled shape. For ancestral configurations at the root, the maximum increases with $k_0^n$, where $k_0 approx 1.5028$ is a quadratic recurrence constant. Under a uniform distribution over the set of labeled trees of given size, the mean number of root ancestral configurations grows with $sqrt{3/2}(4/3)^n$ and the variance with approximately $1.4048(1.8215)^n$. The results provide a contribution to the combinatorial study of gene trees and species trees.
We present a computational model to reconstruct trees of ancestors for animals with sexual reproduction. Through a recursive algorithm combined with a random number generator, it is possible to reproduce the number of ancestors for each generation and use it to constraint the maximum number of the following generation. This new model allows to consider the reproductive preferences of particular species and combine several trees to simulate the behavior of a population. It is also possible to obtain a description analytically, considering the simulation as a theoretical stochastic process. Such process can be generalized in order to use an algorithm associated with it to simulate other similar processes of stochastic nature. The simulation is based in the theoretical model previously presented before.
2-colored best match graphs (2-BMGs) form a subclass of sink-free bi-transitive graphs that appears in phylogenetic combinatorics. There, 2-BMGs describe evolutionarily most closely related genes between a pair of species. They are explained by a unique least resolved tree (LRT). Introducing the concept of support vertices we derive an $O(|V|+|E|log^2|V|)$-time algorithm to recognize 2-BMGs and to construct its LRT. The approach can be extended to also recognize binary-explainable 2-BMGs with the same complexity. An empirical comparison emphasizes the efficiency of the new algorithm.
Least squares trees, multi-dimensional scaling and Neighbor Nets are all different and popular ways of visualizing multi-dimensional data. The method of flexi-Weighted Least Squares (fWLS) is a powerful method of fitting phylogenetic trees, when the exact form of errors is unknown. Here, both polynomial and exponential weights are used to model errors. The exact same models are implemented for multi-dimensional scaling to yield flexi-Weighted MDS, including as special cases methods such as the Sammon Stress function. Here we apply all these methods to population genetic data looking at the relationships of Abrahams Children encompassing Arabs and now widely dispersed populations of Jews, in relation to an African outgroup and a variety of European populations. Trees, MDS and Neighbor Nets of this data are compared within a common likelihood framework and the strengths and weaknesses of each method are explored. Because the errors in this type of data can be complex, for example, due to unexpected genetic transfer, we use a residual resampling method to assess the robustness of trees and the Neighbor Net. Despite the Neighbor Net fitting best by all criteria except BIC, its structure is ill defined following residual resampling. In contrast, fWLS trees are favored by BIC and retain considerable strong internal structure following residual resampling. This structure clearly separates various European and Middle Eastern populations, yet it is clear all of the models have errors much larger than expected by sampling variance alone.
Consensus methods are widely used for combining phylogenetic trees into a single estimate of the evolutionary tree for a group of species. As more taxa are added, the new source trees may begin to tell a different evolutionary story when restricted to the original set of taxa. However, if the new trees, restricted to the original set of taxa, were to agree exactly with the earlier trees, then we might hope that their consensus would either agree with or resolve the original consensus tree. In this paper, we ask under what conditions consensus methods exist that are future proof in this sense. While we show that some methods (e.g. Adams consensus) have this property for specific types of input, we also establish a rather surprising `no-go theorem: there is no reasonable consensus method that satisfies the future-proofing property in general. We then investigate a second notion of future proofing for consensus methods, in which trees (rather than taxa) are added, and establish some positive and negative results. We end with some questions for future work.
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