Bounds on the Expected Size of the Maximum Agreement Subtree

Bernstein, Daniel Irving; Ho, Lam Si Tung; Long, Colby; Steel, Mike; John, Katherine St.; Sullivant, Seth

doi:10.1137/140997750

Cited by 13 publications

(20 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bernstein, Ho, Long, Steel, St. John, and Sullivant [6] established a qualitatively similar upper bound O(n 1/2 ) for the likely size of a common induced subtree in a harder case of Yule-Harding tree, again relying on sampling consistency of this tree model. Recently Misra and Sullivant [19] were able to prove the estimate Θ(n 1/2 ) for the case when two independent binary trees with n labelled leaves are obtained by selecting independently, and uniformly at random, two leaf-labelings of the same unlabelled tree.…”

Section: Introduction Resultsmentioning

confidence: 81%

“…Recently Misra and Sullivant [19] were able to prove the estimate Θ(n 1/2 ) for the case when two independent binary trees with n labelled leaves are obtained by selecting independently, and uniformly at random, two leaf-labelings of the same unlabelled tree. Using the classic results on the length of the longest increasing subsequence in the uniformly random permutation, the authors of [6] established a first power-law lower bound cn 1/8 for the likely size of the common induced subtree in the case of the uniform rooted binary tree, and a lower bound cn a−o (1) , a = 0.344 . .…”

Section: Introduction Resultsmentioning

confidence: 99%

“…Let X n,a denote the random total number of leaf-sets S ⊂ [n] of cardinality a that induce the same rooted subtree in T ′ n and T ′′ n . Bernstein et al [6] proved that…”

Section: Uniform Binary Treesmentioning

confidence: 99%

“…I owe a debt of genuine gratitude to Ovidiu Costin and Jeff McNeal for guiding me to the Weierstrass separation theorem. I thank Daniel Bernstein, Mike Steel, and Seth Sullivant for an important feedback regarding the references [7], [6] and [22].…”

mentioning

confidence: 99%

See 3 more Smart Citations

Expected number of induced subtrees shared by two independent copies of a random tree

Pittel¹

2021

Preprint

View full text Add to dashboard Cite

Consider a rooted tree T of minimum degree 2 at least, with leaf-set [n]. A rooted tree T with leaf-set S ⊂ [n] is induced by S in T if T is the lowest common ancestor subtree for S, with all its degree-2 vertices suppressed. A "maximum agreement subtree" (MAST) for a pair of two trees T ′ and T ′′ is a tree T with a largest leaf-set S ⊂ [n] such that T is induced by S both in T ′ and T ′′ . Bryant et al. [7] and Bernstein et al.[6] proved, among other results, that for T ′ and T ′′ being two independent copies of a random binary (uniform or Yule-Harding distributed) tree T , the likely magnitude order of MAST(T ′ , T ′′ ) is O(n 1/2 ). In this paper we prove this bound for a wide class of random rooted trees: T is a terminal tree of a branching process with an offspring distribution of mean 1, conditioned on "total number of leaves is n".

show abstract

Section: Introduction Resultsmentioning

confidence: 81%

Section: Introduction Resultsmentioning

confidence: 99%

“…Let X n,a denote the random total number of leaf-sets S ⊂ [n] of cardinality a that induce the same rooted subtree in T ′ n and T ′′ n . Bernstein et al [6] proved that…”

Section: Uniform Binary Treesmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Expected number of induced subtrees shared by two independent copies of a random tree

Pittel¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Upper bounds of 𝑂(√𝑛) hold for both standard models in phylogenetics, the uniform model and the Yule-Harding model. On the other hand, Bernstein, Ho, Long, Steel, St. John, and Sullivant [1] provided lower bounds of Ω(𝑛 1/8 ) (uniform model) and Ω(𝑛 0.344 ) (Yule-Harding). Simulations suggest that the true exponent is close to 1 2 ; see [2].…”

Section: Trees In Biologymentioning

confidence: 99%

Trees in Many Contexts

Smith¹,

Bóna²,

Czabarka³

et al. 2022

Notices Amer. Math. Soc.

View full text Add to dashboard Cite

Counting Markov equivalence classes for DAG models on trees

Radhakrishnan

Solus

Uhler

2018

Discrete Applied Mathematics

View full text Add to dashboard Cite

DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause-effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e. skeleton) and the same set of the induced subDAGs i → j ← k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, the authors introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and they studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some important families of graphs including paths, stars, cycles, spider graphs, caterpillars, and complete binary trees. In doing so, we recover important connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictates the number and size of MECs.

show abstract

Bounds on the Expected Size of the Maximum Agreement Subtree

Cited by 13 publications

References 14 publications

Expected number of induced subtrees shared by two independent copies of a random tree

Expected number of induced subtrees shared by two independent copies of a random tree

Trees in Many Contexts

Counting Markov equivalence classes for DAG models on trees

Contact Info

Product

Resources

About