Exact formulas for the variance of several balance indices under the Yule model

Cardona, Gabriel; Mir, Arnau; Rosselló, Francesc

doi:10.1007/s00285-012-0615-9

Cited by 25 publications

(35 citation statements)

References 27 publications

(33 reference statements)

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“…• A tree of shape Q * 3 can only be obtained by adding the leaf 4 to the arc from the root to its only leaf child in some tree K Since E Y (QIB 1 ) = 0 and f QIB (n − 1, 1) = 0, applying the aforementioned result from [4] we have that Dividing by n both sides of this expression for E Y (QIB 2 n ) and setting y n = E Y (QIB 2 n )/n, we obtain the recurrence y n = y n−1 + (n − 2)(n − 3)(253n 4 − 2014n 3 + 6119n 2 − 7430n + 3504) 181440…”

Section: Discussionmentioning

confidence: 99%

A balance index for phylogenetic trees based on quartets

Coronado

Mir

Rosselló

et al. 2018

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We define a new balance index for phylogenetic trees based on the symmetry of the evolutive history of every quartet of leaves. This index makes sense for polytomic trees and it can be computed in time linear in the number of leaves. We compute its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance for bifurcating trees under Ford's α-model and Aldous' β-model and for arbitrary trees under the α-γ-model.

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Section: Discussionmentioning

confidence: 99%

A balance index for phylogenetic trees based on quartets

Coronado

Mir

Rosselló

et al. 2018

Preprint

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“…Value rQI vs S on BT20 −0.889 rQI vs C on BT20 −0.893 rQI vs Φ on BT20 −0.935 rQI vs Ch on BT20 0.165 rQI vs S on T15 −0.787 rQI vs Φ on T15 −0.827 Since E Y (rQIB 1 ) = 0 and f rQIB (n − 1, 1) = 0, applying the aforementioned result from [5] we have that Dividing by n both sides of this expression for E Y (rQIB 2 n ) and setting y n = E Y (rQIB 2 n )/n, we obtain the recurrence y n = y n−1 + (n − 2)(n − 3)(253n 4 − 2014n 3 + 6119n 2 − 7430n + 3504) 181440 .…”

Section: Correlationmentioning

confidence: 98%

A balance index for phylogenetic trees based on rooted quartets

et al. 2019

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We define a new balance index for rooted phylogenetic trees based on the symmetry of the evolutive history of every set of 4 leaves. This index makes sense for multifurcating trees and it can be computed in time linear in the number of leaves. We determine its maximum and minimum values for arbitrary and bifurcating trees, and we provide exact formulas for its expected value and variance on bifurcating trees under Ford's α-model and Aldous' β-model and on arbitrary trees under the α-γ-model. of isomorphisms of the restriction of T to them (the rooted quartet they define), and then we add up these values over all 4-tuples of different leaves of T . The idea behind the definition of this balance index is that a highly symmetrical evolutive process should give rise to symmetrical evolutive histories of many small subsets of taxa. In terms of phylogenetic trees, this leads us to expect that, the most symmetrical a phylogenetic tree is, the most symmetrical will be its restrictions to subsets of leaves of a fixed cardinality. Since the smallest number of leaves yielding enough different tree topologies to allow a meaningful comparison of their symmetry is 4, we assess the balance of a tree by measuring the symmetry of all its rooted quartets and adding up the results. And indeed, in Section 4 below we shall find the trees with maximum and minimum values of our rooted quartet index in both the arbitrary and the bifurcating cases, and it will turn out that the minimum value is reached exactly at the combs (see Fig. 1.(a)), which are usually considered the least balanced trees, and the maximum value is reached, in the arbitrary case, exactly at the rooted stars (see Fig. 1.(b)) and, in the bifurcating case, exactly at the maximally balanced trees (cf. Fig. 3 ), which in both cases are considered the most balanced trees.Besides taking its maximum and minimum values at the expected trees, other important features of our index are that it can be easily computed in linear time and that its mean value and variance can be explicitly computed on any probabilistic model of phylogenetic trees satisfying two natural conditions: independence under relabelings and sampling consistency. This allows us to provide these values for two well-known probabilistic models of bifurcating phylogenetic trees, Ford's α-model [13] and Aldous' β-model [2], which include as specific instances the Yule [14,29] and the uniform [6,24,19] models, as well as for Chen-Ford-Winkel's α-γ-model of multifurcating trees [7]. To our knowledge, this is the first shape index for which closed formulas for the expected value and the variance under the α-γ-model have been provided.The rest of this paper is organized as follows. In the next section we introduce the basic notations and facts on phylogenetic trees that will be used in the rest of the paper, and we recall several preliminary results on probabilistic models of phylogenetic trees, proving those results for which we have not been able to find a suitable reference in the literature. Then, in Section 3, we ...

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“…3.1) Since QB(10) = {(5, 5), (6, 4)}, the different ways of splitting the leaves of the tree (10, 10) produce the trees ((5, 5), (5, 5)), ((5, 5), (6,4)), and ((6, 4), (6,4)). Now, since QB(5) = {(3, 2)}, QB(6) = {(3, 3), (4, 2)}, and QB(3) = {(2, 1)}, and 1, 2, and 4 are powers of 2, we have the following derivations from these trees through all possible combinations of splitting the leaves in the trees:…”

Section: Characterizing and Generating Minimal Colless Treesmentioning

confidence: 99%

“…((5, 5), (5,5)) ⇒ (((3, 2), (3, 2)), ((3, 2), (3, 2))) ⇒ ((((2, 1), 2), ((2, 1), 2)), (((2, 1), 2), ((2, 1), 2))) ⇒ (((((·, ·), ·), (·, ·)), (((·, ·), ·), (·, ·))), ((((·, ·), ·), (·, ·)), (((·, ·), ·), (·, ·)))) ((5, 5), (6,4)) ⇒ (((3, 2), (3, 2)), ((3, 3), 4)) ⇒ ((((2, 1), 2), ((2, 1), 2)), (((2, 1), (2, 1)), 4)) ⇒ (((((·, ·), ·), (·, ·)), (((·, ·), ·), (·, ·))), ((((·, ·), ·), ((·, ·), ·)), ((·, ·), (·, ·))) ((5, 5), (6,4)) ⇒ (((3, 2), (3, 2)), ((4, 2), 4)) ⇒ ((((2, 1), 2), ((2, 1), 2)), ((4, 2), 4)) ⇒ (((((·, ·), ·), (·, ·)), (((·, ·), ·), (·, ·))), ((((·, ·), (·, ·)), (·, ·)), ((·, ·), (·, ·)))) ((6, 4), (6,4)) ⇒ (((3, 3), 4), ((3, 3), 4)) ⇒ ((((2, 1), (2, 1)), 4), (((2, 1), (2, 1)), 4)) ⇒ (((((·, ·), ·), ((·, ·), ·)), ((·, ·), (·, ·))), ((((·, ·), ·), ((·, ·), ·)), ((·, ·), (·, ·)))) ((6, 4), (6,4)) ⇒ (((3, 3), 4), ((4, 2), 4)) ⇒ ((((2, 1), (2, 1)), 4), ((4, 2), 4)) ⇒ (((((·, ·), ·), ((·, ·), ·)), ((·, ·), (·, ·))), ((((·, ·), (·, ·)), (·, ·)), ((·, ·), (·, ·)))) ((6, 4), (6,4)) ⇒ (((4, 2), 4), ((4, 2), 4)) ⇒ (((((·, ·), (·, ·)), (·, ·)), ((·, ·), (·, ·))), ((((·, ·), (·, ·)), (·, ·)), ((·, ·), (·, ·)))) (2, 1), (2, 1)), ((2, 1), (2, 1))), 8) ⇒ (((((·, ·), ·), ((·, ·), ·)), (((·, ·), ·), ((·, ·), ·))), (((·, ·), (·, ·)), ((·, ·), (·, ·)))) ((6, 6), 8) ⇒ (((3, 3), (4, 2)), 8) ⇒ ((((2, 1), (2, 1)), (4, 2)), 8) ⇒ (((((·, ·), ·), ((·, ·), ·)), (((·, ·), (·, ·)), (·, ·))), (((·, ·), (·, ·)), ((·, ·), (·, ·)))) ((6, 6), 8) ⇒ (((4, 2), (4, 2)), 8) ⇒ (((((·, ·), (·, ·)), (·, ·)), (((·, ·), (·, ·)), (·, ·))), (((·, ·), (·, ·)), ((·, ·), (·, ·)))) ((8, 4), 8) ⇒ (((((·, ·), (·, ·)), ((·, ·), (·, ·))), ((·, ·), (·, ·))), (((·, ·), (·, ·)), ((·, ·), (·, ·)))) So, there are 10 different minimal Colless trees in T 20 . We depict them in Figure 5 below.…”

Section: Characterizing and Generating Minimal Colless Treesunclassified

“…Among them, the Colless index, introduced by Colless [8], is one of the oldest and most popular. It is defined, on a rooted bifurcating tree T , as the sum, over all the internal nodes v of T , of the absolute value of the difference between the numbers of descendant leaves of the pair of children of v. Its statistical properties under several probabilistic models for phylogenetic trees have been thoroughly studied: see, for instance, [5,6,14,16].…”

Section: Introductionmentioning

confidence: 99%

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On the minimum value of the Colless index and the bifurcating trees that achieve it

et al. 2020

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Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless [8]. While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with n leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the trees that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

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Exact formulas for the variance of several balance indices under the Yule model

Cited by 25 publications

References 27 publications

A balance index for phylogenetic trees based on quartets

A balance index for phylogenetic trees based on quartets

A balance index for phylogenetic trees based on rooted quartets

On the minimum value of the Colless index and the bifurcating trees that achieve it

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