Testing the agreement of trees with internal labels

Background A semi-labeled tree is a tree where all leaves as well as, possibly, some internal nodes are labeled with taxa. Semi-labeled trees encompass ordinary phylogenetic trees and taxonomies. Suppose we are given a collection $${\mathcal {P}}= \{{\mathcal {T}}_1, {\mathcal {T}}_2, \ldots , {\mathcal {T}}_k\}$$ P = { T 1 , T 2 , … , T k } of semi-labeled trees, called input trees, over partially overlapping sets of taxa. The agreement problem asks whether there exists a tree $${\mathcal {T}}$$ T , called an agreement tree, whose taxon set is the union of the taxon sets of the input trees such that the restriction of $${\mathcal {T}}$$ T to the taxon set of $${\mathcal {T}}_i$$ T i is isomorphic to $${\mathcal {T}}_i$$ T i , for each $$i \in \{1, 2, \ldots , k\}$$ i ∈ { 1 , 2 , … , k } . The agreement problems is a special case of the supertree problem, the problem of synthesizing a collection of phylogenetic trees with partially overlapping taxon sets into a single supertree that represents the information in the input trees. An obstacle to building large phylogenetic supertrees is the limited amount of taxonomic overlap among the phylogenetic studies from which the input trees are obtained. Incorporating taxonomies into supertree analyses can alleviate this issue. Results We give a $${\mathcal {O}}(n k (\sum _{i \in [k]} d_i + \log ^2(nk)))$$ O ( n k ( ∑ i ∈ [ k ] d i + log 2 ( n k ) ) ) algorithm for the agreement problem, where n is the total number of distinct taxa in $${\mathcal {P}}$$ P , k is the number of trees in $${\mathcal {P}}$$ P , and $$d_i$$ d i is the maximum number of children of a node in $${\mathcal {T}}_i$$ T i . Conclusion Our algorithm can aid in integrating taxonomies into supertree analyses. Our computational experience with the algorithm suggests that its performance in practice is much better than its worst-case bound indicates.

Testing the Agreement of Trees with Internal Labels

Background: A semi-labeled tree is a tree where all leaves as well as, possibly, some internal nodes are labeledwith taxa. Semi-labeled trees encompass ordinary phylogenetic trees and taxonomies. Suppose we are given a collection P = {T1, T2, . . . , Tk} of semi-labeled trees, called input trees, over partially overlapping sets of taxa. The agreement problem asks whether there exists a tree T , called an agreement tree, whose taxon set is the union of the taxon sets of the input trees such that the restriction of T to the taxon set of Ti is isomorphic to i, for each i ε 1, 2, . . . , k . The agreement problems is a special case of the supertree problem, the problem of synthesizing a collection of phylogenetic trees with partially overlapping taxon sets into a single supertree that represents the information in the input trees. An obstacle to building large phylogenetic supertrees is the limited amount of taxonomic overlap among the phylogenetic studies from which the input trees are obtained. Incorporating taxonomies into supertree analyses can alleviate this issue. Results: We give a O(nk(i∈[k]di + log2(nk))) algorithm for the agreement problem, where n is the total number of distinct taxa in P, k is the number of trees in P, and di is the maximum number of children of a node in Ti. Our computational experience with the algorithm suggests that its performance in practice is much better than its worst-case bound indicates.

Accessibility percolation on random rooted labeled trees

The accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.

Torsion-free abelian groups with optimal Scott families

We prove that for any computable successor ordinal of the form [Formula: see text] [Formula: see text] limit and [Formula: see text] there exists computable torsion-free abelian group [Formula: see text]TFAG[Formula: see text] that is relatively [Formula: see text] -categorical and not [Formula: see text] -categorical. Equivalently, for any such [Formula: see text] there exists a computable TFAG whose initial segments are uniformly described by [Formula: see text] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [Formula: see text]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) [Formula: see text]-categorical TFAGs for arbitrarily large [Formula: see text] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [Formula: see text]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.