On the Combinatorics of Rooted Binary Phylogenetic Trees

We study subtree-prune-and-regraft (SPR) operations on leaf-labelled rooted binary trees, also known as rooted binary phylogenetic trees. This study is motivated by the problem of graphically representing evolutionary histories of biological sequences subject to recombination. We investigate some basic properties of the induced SPR-metric on the space of leaf-labelled rooted binary trees with n leaves. In contrast to the case of unrooted trees, the number |U(T)| of trees in which are one SPR operation away from a given tree depends on the topology of T. In this paper, we construct recursion relations which allow one to determine the unit-neighbourhood size |U(T)| efficiently for any tree topology. In fact, using the recursion relations we are able to derive a simple closed-form formula for the unit-neighbourhood size. As a corollary, we construct sharp upper and lower bounds on the size of unit-neighbourhoods and investigate the diameter of . Lastly, we consider an enumeration problem relevant to population genetics.AMS Subject Classification: 05C05, 92D15.     

Properties of Subtree-Prune-and-Regraft Operations on Totally-Ordered Phylogenetic Trees

We study some properties of subtree-prune-and-regraft (SPR) operations on leaflabelled rooted binary trees in which internal vertices are totally ordered. Since biological events occur with certain time ordering, sometimes such totally-ordered trees must be used to avoid possible contradictions in representing evolutionary histories of biological sequences. Compared to the case of plain leaf-labelled rooted binary trees where internal vertices are only partially ordered, SPR operations on totally-ordered trees are more constrained and therefore more difficult to study. In this paper, we investigate the unit-neighbourhood U(T), defined as the set of totally-ordered trees one SPR operation away from a given totally-ordered tree T. We construct a recursion relation for | U(T) | and thereby arrive at an efficient method of determining | U(T) |. In contrast to the case of plain rooted trees, where the unit-neighbourhood size grows quadratically with respect to the number n of leaves, for totally-ordered trees | U(T) | grows like O(n³). For some special topology types, we are able to obtain simple closed-form formulae for | U(T) |. Using these results, we find a sharp upper bound on | U(T) | and conjecture a formula for a sharp lower bound. Lastly, we study the diameter of the space of totally-ordered trees measured using the induced SPR-metric. Received May 18, 2004     

On agreement forests

Tree rearrangement operations are widely used to measure the dissimilarity between phylogenetic trees with identical leaf sets. The tree bisection and reconnection (tbr) distance for unrooted trees can be equivalently defined in terms of agreement forests. For both the tbr distance and the less general subtree prune and regraft (spr) distance, we use such forests to derive new upper and lower bounds on the maximal possible distance between two trees with n leaves.     

The distribution of height and diameter in random non‐plane binary trees

This study is dedicated to precise distributional analyses of the height of non‐plane unlabelled binary trees (“Otter trees”), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size n is proved to admit a limiting theta distribution, both in a central and local sense, and obey moderate as well as large deviations estimates. The approximations obtained for height also yield the limiting distribution of the diameter of unrooted trees. The proofs rely on a precise analysis, in the complex plane and near singularities, of generating functions associated with trees of bounded height. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel��s conjecture

A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on n leaves, sample sequences of length ??(log n) are needed. It was conjectured by Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than ${p^{\ast} = (\sqrt{2}-1)/2^{3/2}}$ , then the tree can be recovered from sequences of length O(log n). The value p* is given by the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel??s conjecture was proven by the second author in the special case where the tree is ??balanced.?? The second author also proved that if all edges have mutation probability larger than p* then the length needed is n ^??(1). Here we show that Steel??s conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from O(log n)-length sequences when the mutation probabilities are discretized and less than p*. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.     

On the deque conjecture for the splay algorithm

Splay is a simple, efficient algorithm for searching binary search trees, devised by Sleator and Tarjan, that reorganizes the tree after each search by means of rotations. An open conjecture of Sleator and Tarjan states that Splay is, in essence, the fastest algorithm for processing any sequence of search operations on a binary search tree, using only rotations to reorganize the tree. Tarjan proved a basic special case of this conjecture, called theScanning Theorem, and conjectured a more general special case, called theDeque Conjecture. The Deque Conjecture states that Splay requires linear time to process sequences of deque operations on a binary tree. We prove the following results:

1. Almost tight lower and upper bounds on the maximum numbers of occurrences of various types of right rotations in a sequence of right rotations performed on a binary tree. In particular, the lower bound for right 2-turns refutes Sleator's Right Turn Conjecture.
2. A linear times inverse Ackerman upper bound for the Deque Conjecture. This bound is derived using the above upper bounds.
3. Two new proofs of the Scanning Theorem, one, a simple potential-based proof that solves Tarjan's problem of finding a potential-based proof for the theorem, the other, an inductive proof that generalizes the theorem.

     

The continuum random tree is the scaling limit of unlabeled unrooted trees

We show that the uniform unlabeled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov‐Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Pólya trees to unrooted trees.     

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set {1,2,…,n}, the local direction of each edge (ij) is from i to j if i<j. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ=e₁¹e₂²… of a tree on the vertex set {1,2,…,n} is a partition of n−1. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.     

A note on the cubical dimension of new classes of binary trees

The cubical dimension of a graph G is the smallest dimension of a hypercube into which G is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with 2 ⁿ vertices, n ? 1, is n. The 2-rooted complete binary tree of depth n is obtained from two copies of the complete binary tree of depth n by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.     

On Rotations and the Generation of Binary Trees

The rotation graph, G_n, has vertex set consisting of all binary trees with n nodes. Two vertices are connected by an edge if a single rotation will transform one tree into the other. We provide a simpler proof of a result of Lucas that G_n, contains a Hamilton path. Our proof deals directly with the pointer representation of the binary tree. This proof provides the basis of an algorithm for generating all binary trees that can be implemented to run on a pointer machine and to use only constant time between the output of successive trees. Ranking and unranking algorithms are developed for the ordering of binary trees implied by the generation algorithm. These algorithms have time complexity O(n²) (arithmetic operations). We also show strong relationships amongst various representations of binary trees and amongst binary tree generation algorithms that have recently appeared in the literature.     

Maximal Flow in Branching Trees and Binary Search Trees

A capacitated network is a tree with a non negative number, called capacity, associated to each edge. The maximal flow that can pass through a given path is the minimun capacity on the path. Antal and Krapivski (Phys Rev E 74:051110, 2006) study the distribution for the maximal flow from the root to a leaf in the case of a deterministic binary tree with independent and identically distributed random capacities. In this paper their result is extended to three classes of trees with a random number of children and dependent random capacities: binary trees with general capacities distribution, branching trees with exchangeable capacities and random binary search trees.     

Approximation algorithms for constrained generalized tree alignment problem

In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic tree for S is known. In such instances, we would like to make use of this knowledge to restrict the tree topologies that we consider and construct a biologically relevant minimum cost evolutionary tree. So, we propose the following natural generalization of the generalized tree alignment problem, a problem known to be MAX-SNP Hard, stated as follows:

Constrained Generalized Tree Alignment Problem [S. Divakaran, Algorithms and heuristics for constrained generalized alignment problem, DIMACS Technical Report 2007-21, 2007]: Given a set S of k related sequences and a phylogenetic forest comprising of node-disjoint phylogenetic trees that specify the topological constraints that an evolutionary tree of S needs to satisfy, construct a minimum cost evolutionary tree for S.

In this paper, we present constant approximation algorithms for the constrained generalized tree alignment problem. For the generalized tree alignment problem, a special case of this problem, our algorithms provide a guaranteed error bound of 2−2/k.     

On the approximability of the Maximum Agreement SubTree and Maximum Compatible Tree problems

The aim of this paper is to give a complete picture of approximability for two tree consensus problems which are of particular interest in computational biology: Maximum Agreement SubTree (MAST) and Maximum Compatible Tree (MCT). Both problems take as input a label set and a collection of trees whose leaf sets are each bijectively labeled with the label set. Define the size of a tree as the number of its leaves. The well-known MAST problem consists of finding a maximum-sized tree that is topologically embedded in each input tree, under label-preserving embeddings. Its variant MCT is less stringent, as it allows the input trees to be arbitrarily refined. Our results are as follows. We show that MCT is -hard to approximate within bound n^1−? on rooted trees, where n denotes the size of each input tree; the same approximation lower bound was already known for MAST [J. Jansson, Consensus algorithms for trees and strings, Ph. D. Thesis, Lund University, 2003]. Furthermore, we prove that MCT on two rooted trees is not approximable within bound 2^log1−?n, unless all problems in are solvable in quasi-polynomial time; the same result was previously established for MAST on three rooted trees [J. Hein, T. Jiang, L. Wang, K. Zhang, On the complexity of comparing evolutionary trees, Discrete Applied Mathematics 71 (1-3) (1996) 153-169] (note that MAST on two trees is solvable in polynomial time [M.A. Steel, T.J. Warnow, Kaikoura tree theorems: Computing the maximum agreement subtree, Information Processing Letters 48 (2) (1993) 77-82]). Let CMAST, resp. CMCT, denote the complement version of MAST, resp. MCT: CMAST, resp. CMCT, consists of finding a tree that is a feasible solution of MAST, resp. MCT, and whose leaf label set excludes a smallest subset of the input labels. The approximation threshold for CMAST, resp. CMCT, on rooted trees is shown to be the same as the approximation threshold for CMAST, resp. CMCT, on unrooted trees; it was already known that both CMAST and CMCT are approximable within ratio three on rooted and unrooted trees [V. Berry, F. Nicolas, Maximum agreement and compatible supertrees, in: S.C. Sahinalp, S. Muthukrishnan, U. Dogrusoz (Eds.), Proceedings of the 15th Annual Symposium on Combinatorial Pattern Matching, CPM’04, in: LNCS, vol. 3109, Springer-Verlag, 2004, pp. 205-219; G. Ganapathy, T.J. Warnow, Approximating the complement of the maximum compatible subset of leaves of k trees, in: K. Jansen, S. Leonardi, V. V. Vazirani (Eds.), Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX’02, in: LNCS, vol. 2462, Springer-Verlag, 2002, pp. 122-134]. The latter results are completed by showing that CMAST is -hard on three rooted trees and that CMCT is -hard on two rooted trees.     

Slim sets of binary trees

A classical problem in phylogenetic tree analysis is to decide whether there is a phylogenetic tree T that contains all information of a given collection P of phylogenetic trees. If the answer is “yes” we say that P is compatible and T displays P. This decision problem is NP-complete even if all input trees are quartets, that is binary trees with exactly four leaves. In this paper, we prove a sufficient condition for a set of binary phylogenetic trees to be compatible. That result is used to give a short and self-contained proof of the known characterization of quartet sets of minimal cardinality which are displayed by a unique phylogenetic tree.     

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.     

Tutte polynomials for trees

We define two two-variable polynomials for rooted trees and one two-variable polynomial for unrooted trees, all of which are based on the coranknullity formulation of the Tutte polynomial of a graph or matroid. For the rooted polynomials, we show that the polynomial completely determines the rooted tree, i.e., rooted trees T₁ and T₂ are isomorphic if and only if f(T₁) = f(T₂). The corresponding question is open in the unrooted case, although we can reconstruct the degree sequence, number of subtrees of size k for all k, and the number of paths of length k for all k from the (unrooted) polynomial. The key difference between these three polynomials and the standard Tutte polynomial is the rank function used; we use pruning and branching ranks to define the polynomials. We also give a subtree expansion of the polynomials and a deletion-contraction recursion they satisfy.     

Twist–Rotation Transformations of Binary Trees and Arithmetic Expressions

The paper studies the computational complexity and efficient algorithms for the twist–rotation transformations of binary trees, which is equivalent to the transformation of arithmetic expressions over an associative and commutative binary operation. The main results are (1) a full binary tree with n labeled leaves can be transformed into any other in at most 3n log n + 2n twist and rotation operations, (2) deciding the twist–rotation distance between two binary trees is NP-complete, and (3) the twist–rotation transformation can be approximated with ratio 6 log n + 4 in polynomial time for full binary trees with n uniquely labeled leaves.     

The Splits in the Neighborhood of a Tree

A phylogenetic tree represents historical evolutionary relationships between different species or organisms. The space of possible phylogenetic trees is both complex and exponentially large. Here we study combinatorial features of neighbourhoods within this space, with respect to four standard tree metrics. We focus on the splits of a tree: the bipartitions induced by removing a single edge from the tree. We characterize those splits appearing in trees that are within a given distance of the original tree, demonstrating close connections between these splits, the Whitney number of a tree, and the binary characters with a given parsimony length.AMS Subject Classification: 68R10, 05C05, 68Q25, 92D15.