Reconstruction of Rooted Directed Trees

Let T be a rooted directed tree on n vertices, rooted at v. The rooted subtree frequency vector (RSTF-vector) of T with root v, denoted by rstf(T, v) is a vector of length n whose entry at position k is the number of subtrees of T that contain v and have exactly k vertices. In this paper we present an algorithm for reconstructing rooted directed trees from their rooted subtree frequencies (up to isomorphism). We show that there are examples of nonisomorphic pairs of rooted directed trees that are RSTF-equivalent, that is they share the same rooted subtree frequency vectors. We have found all such pairs (groups) for small sizes by using exhaustive computer search. We show that infinitely many nonisomorphic RSTF-equivalent pairs of trees exist by constructing infinite families of examples

Reliability analysis of reconstructing phylogenies under long branch attraction conditions

Master's Project (M.S.) University of Alaska Fairbanks, 2018.In this simulation study we examined the reliability of three phylogenetic reconstruction techniques in a long branch attraction (LBA) situation: Maximum Parsimony (M P), Neighbor Joining (NJ), and Maximum Likelihood. Data were simulated under five DNA substitution models-JC, K2P, F81, HKY, and G T R-from four different taxa. Two branch length parameters of four taxon trees ranging from 0.05 to 0.75 with an increment of 0.02 were used to simulate DNA data under each model. For each model we simulated DNA sequences with 100, 250, 500 and 1000 sites with 100 replicates. When we have enough data the maximum likelihood technique is the most reliable of the three methods examined in this study for reconstructing phylogenies under LBA conditions. We also find that MP is the most sensitive to LBA conditions and that Neighbor Joining performs well under LBA conditions compared to MP

Phase transitions in Phylogeny

We apply the theory of markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least 3, and the net transition on each edge is bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2 e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree, having n leaves, can be reconstructed from O(log n) samples (characters) at the leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then there exist topologies which require at least poly(n) samples for reconstruction. Our results are the first rigorous results to establish the role of phase transitions for markov random fields on trees as studied in probability, statistical physics and information theory to the study of phylogenies in mathematical biology.Comment: To appear in Transactions of the AM

An Arrow-type result for inferring a species tree from gene trees

The reconstruction of a central tendency `species tree' from a large number of conflicting gene trees is a central problem in systematic biology. Moreover, it becomes particularly problematic when taxon coverage is patchy, so that not all taxa are present in every gene tree. Here, we list four desirable properties that a method for estimating a species tree from gene trees should have. We show that while these can be achieved when taxon coverage is complete (by the Adams consensus method), they cannot all be satisfied in the more general setting of partial taxon coverage.Comment: 5 pages, 0 figure