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arXiv:0912.4472v2 [q-bio.PE] 29 Jul 2010
Noname manuscript No.
(will be inserted by the editor)
Identifying the Rooted Species Tree from the Distribution
of Unrooted Gene Trees under the Coalescent
Elizabeth S. Allman · James H. Degnan ·
John A. Rhodes
Received: date / Accepted: date
July 30, 2010
Abstract Gene trees are evolutionary trees representing the ancestry of genes sam-
pled from multiple populations. Species trees represent populations of individuals —
each with many genes — splitting into new populations or species. The coalescent pro-
cess, which models ancestry of gene copies within populations, is often used to model
the probability distribution of gene trees given a fixed species tree. This multispecies
coalescent model provides a framework for phylogeneticists to infer species trees from
gene trees using maximum likelihood or Bayesian approaches. Because the coalescent
models a branching process over time, all trees are typically assumed to be rooted in
this setting. Often, however, gene trees inferred by traditional phylogenetic methods
are unrooted.
We investigate probabilities of unrooted gene trees under the multispecies coa-
lescent model. We show that when there are four species with one gene sampled per
species, the distribution of unrooted gene tree topologies identifies the unrooted species
tree topology and some, but not all, information in the species tree edges (branch
lengths). The location of the root on the species tree is not identifiable in this sit-
uation. However, for 5 or more species with one gene sampled per species, we show
that the distribution of unrooted gene tree topologies identifies the rooted species tree
topology and all its internal branch lengths. The length of any pendant branch leading
E. S. Allman
Department of Mathematics and Statistics, University of Alaska Fairbanks,
PO Box 756660, Fairbanks, AK 99775 USA
E-mail: e.allman@alaska.edu
Corresponding author:
J. H. Degnan
Department of Mathematics and Statistics, University of Canterbury
Private Bag 4800, Christchurch, New Zealand
E-mail: J.Degnan@math.canterbury.ac.nz
J. A. Rhodes
Department of Mathematics and Statistics, University of Alaska Fairbanks,
PO Box 756660, Fairbanks, AK 99775 USA
E-mail: j.rhodes@alaska.edu
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to a leaf of the species tree is also identifiable for any species from which more than
one gene is sampled.
Keywords Multispecies coalescent · phylogenetics · invariants · polytomy
Mathematics Subject Classification (2000) 62P10 · 92D15
1 Introduction
The goal of a phylogenetic study is often to infer an evolutionary tree depicting the
history of speciation events that lead to a currently extant set of taxa. In these species
trees, speciation events are idealized as populations instantaneously diverging into two
populations that no longer exchange genes. Such trees are often estimated indirectly,
from DNA sequences for orthologous genes from the extant species. A common as-
sumption has been that such an inferred gene tree has a high probability of having
the same topology as the species tree. Recently, however, increasing attention has been
given to population genetic issues that lead to differences between gene and species
trees, and how potentially discordant trees for many genes might be utilized in species
tree inference.
Methods that infer gene trees, such as maximum likelihood (ML) using standard
DNA substitution models, typically can estimate the expected number of mutations
on the edges of a tree, but not the direction of time. Phylogenetic methods therefore
often estimate unrooted gene trees. In many cases, the root of a tree can be inferred
by including data on an outgroup, i.e., a species believed to be less closely related
to the species of interest than any of those are to each other (Jennings and Edwards
2005; Poe and Chubb 2004; Rokas et al 2003). However, outgroup species which are too
distantly related to the ingroup taxa may lead to unreliable inference, and in some cases
appropriate outgroup species are not known (Graham et al 2002; Huelsenbeck et al
2002). The root of a gene tree can alternately be inferred under a molecular clock
assumption, i.e., if mutation rates are constant throughout the edges of a tree. In many
empirical studies, however, such a molecular clock assumption is violated. Furthermore,
without a molecular clock, inferred branch lengths on gene trees may not directly reflect
evolutionary time, as substitution rates vary from branch to branch. For these reasons,
one may have more confidence in the inference of unrooted topological gene trees than
in metric and/or rooted versions.
Methods for inferring rooted species trees from multiple genes have been developed
which make use of rooted gene trees, topological or metric, which possibly differ from
that of the species tree. Most commonly, such methods assume that the incongruent
gene trees (i.e., gene trees with topologies different from the species tree) arise because
of incomplete lineage sorting, the phenomenon that the most recent common ancestor
for two gene copies is more ancient than the most recent population ancestral to the
species from which the genes were sampled. Examples are shown in Fig. 1a-g, in which
the lineages sampled from species a and b do not coalesce in the population immediately
ancestral to a and b. Several approaches for inferring species trees in this setting have
been proposed, such as minimizing deep coalesce (Maddison and Knowles 2006), BEST
(Liu and Pearl 2007), ESP (Carstens and Knowles 2007), STEM (Kubatko et al 2009),
Maximum Tree (Liu et al 2010) (also called the GLASS tree (Mossel and Roch 2010)),
and *BEAST (Heled and Drummond 2010). The analysis of incomplete lineage sorting
requires thinking of rooted trees (the idea of an event being “more ancient” requires
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(a)
abcde
(((B,E),A),(C,D))
(b)
abcde
(((C,D),A),(B,E))
(c)
abcde
((((B,E),A),C),D)
(d)
abcde
((((B,E),A),D),C)
(e)
abcde
((((C,D),A),B),E)
(f)
abcde
((((C,D),A),E),B)
(g)
abcde
(((B,E),(C,D)),A)
T15
(h)
B
E
A
C
D
Fig. 1 The unrooted gene tree T15 in the species tree ((((a, b),c),d),e). The seven distinct
rooted gene trees depicted in (a)–(g) all correspond to the same unrooted gene tree T15
shown in (h). The rooted gene trees in (c) and (d) can only occur for this species tree if all
coalescences occur above the root, in the population with the lightest shading. The rooted
gene trees in (a), (b), (e), (f), and (g) can occur with coalescent events either all above the
root or with some event in the population immediately descended from the root. Only one
coalescent scenario is shown for each of the rooted gene trees.
that time have a direction), and is modeled probabilistically using coalescent theory
(Hudson 1983; Kingman 1982; Nordborg 2001; Tajima 1983; Wakeley 2008).
The coalescent process was first developed to model ancestry of genes by a tree
embedded within a single population, and uses exponential waiting times (going back-
wards in time) until two lineages coalesce. By conceptualizing a species tree as a tree
of connected populations (cf. Fig. 1), each with its own coalescent process, the mul-
tispecies coalescent can model probabilities of rooted gene tree topologies within a
rooted species tree (Degnan and Rosenberg 2009; Degnan and Salter 2005; Nei 1987;
Pamilo and Nei 1988; Rosenberg 2002; Takahata 1989). Although much of the work
of this area has focused on one gene lineage sampled per population, extensions to
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computing gene tree probabilities when more than one lineage is sampled from each
population has also been derived (Degnan 2010; Rosenberg 2002; Takahata 1989).
Under the multispecies coalescent, the species tree is a parameter, consisting of a
rooted tree topology with strictly positive edge weights (branch lengths) on all interior
edges. Pendant edge weights are not specified when there is only one gene sampled per
species, because it is not possible for coalescent events to occur on these edges. Rooted
gene tree topologies are treated as a discrete random variable whose distribution is
parameterized by the species tree, with a state space of size (2n − 3)!! = 1 × 3 × ··· ×
(2n − 3), the number of rooted, binary tree topologies (Felsenstein 2004) for n extant
species (leaves). (Nonbinary gene trees are not included in the sample space since the
coalescent model assigns them probability zero.)
Results on rooted triples (rooted topological trees obtained by considering subsets
of three species) imply that the distribution of rooted gene tree topologies identifies
the rooted species tree topology (Degnan et al 2009), in spite of the fact that the most
likely n-taxon gene tree topology need not have the same topology as the species tree
for n > 3 (Degnan and Rosenberg 2006). Internal branch lengths on the species tree
can also be recovered using probabilities of rooted triples from gene trees. In particular,
for a 3-taxon species tree in which two species a and b are more closely related to each
other than to c, let t denote the internal branch length. If p is the known probability
that on a random rooted topological gene tree, genes sampled from species a and b
are more closely related to each other than either is to a gene sampled from c, then
t = −log((3/2)(1 − p)) (Nei 1987; Wakeley 2008). Thus, for each population (edge)
e of the species tree, choosing two leaves whose most recent ancestral population is e
and one leaf descended from the immediate parental node of e, the length of e can be
determined. We summarize these results as:
Proposition 1 For a species tree with n ≥ 3 taxa, the probabilities of rooted triple
gene tree topologies determine the species tree topology and internal branch lengths.
Because the probability of any rooted triple is the probability that a rooted gene
tree displays the triple, we have the following.
Corollary 2 For a species tree with n ≥ 3 taxa, the distribution of gene tree topologies
determines the species tree topology and internal branch lengths.
Although previous work on modeling gene trees under the coalescent has assumed
that trees are rooted, the event that a particular unrooted topological gene tree is
observed can be regarded as the event that any of its rooted versions occurs at that locus
(Heled and Drummond 2010). For n species, there are (2n − 5)!! unrooted gene trees,
and each unrooted gene tree can be realized by 2n−3 rooted gene trees, corresponding
to choices of an edge on which to place the root. The probability of an n-leaf unrooted
gene tree is therefore the sum of 2n−3 rooted gene tree probabilities, and the unrooted
gene tree probabilities form a well-defined probability distribution.
In this paper, we study aspects of the distribution of unrooted topological gene
trees that arises under the multispecies coalescent model on a species tree, with the
goal of understanding what one may hope to infer about the species tree. We find
that when there are only four species, with one lineage sampled from each, the most
likely unrooted gene tree topology has the same unrooted topology as the species tree;
however, it is impossible to recover the rooted topology of the species tree, or all
information about edge weights, from the distribution of gene trees. When there are
5 or more species, the probability distribution on the unrooted gene tree topologies
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identifies the rooted species tree and all internal edge weights. If multiple samples are
taken from one of more species, then those pendant edge weights become identifiable,
and the total number of taxa required for identifying the species tree can be reduced.
In the main text, we derive these results assuming binary — fully resolved — species
trees. However, the results generalize to nonbinary species trees, which have internal
nodes of outdegree greater than or equal to 2. Details for nonbinary cases are given in
Appendix C. Implications for data analysis will be discussed in Section 6.
We briefly indicate our approach. Because the distribution of the (2n−3)!! (rooted)
or (2n − 5)!! (unrooted) gene trees is determined by the species tree topology and its
n−2 internal branch lengths, gene tree distributions are highly constrained under the
multispecies coalescent model. Calculations show that many gene tree probabilities
are necessarily equal, or satisfy more elaborate polynomial constraints. Polynomials in
gene tree probabilities which evaluate to 0 for any set of branch lengths on a particular
species tree topology are called invariants of the gene tree distribution for that species
tree topology. A trivial example, valid for any species tree, is that the sum of all gene
tree probabilities minus 1 equals 0. Many other invariants express ties in gene tree
probabilities. For example, consider the rooted species tree ((a,b),c), where t is the
length of the internal branch. Suppose gene A is sampled from species a, B from b, and
C from c. Then the rooted gene tree ((A,B),C) has probability p1= 1−(2/3)exp(−t)
under the coalescent, while the two alternative gene trees, ((A,C),B) and ((B,C),A),
have probability p2= p3= (1/3)exp(−t) (Nei 1987). Thus a rooted gene tree invariant
for this species tree is
p2− p3= 0. (1)
We emphasize that this invariant holds for all values of the branch length t. The species
tree also implies certain inequalities in the gene tree distribution; for example, for any
branch length t > 0, p1> p2. Because of such inequalities, the invariant in equation (1)
holds on a gene tree distribution if, and only if, the species tree has topology ((a,b),c).
Different species tree topologies imply different sets of invariants and inequalities
for their gene tree distributions, for both rooted and unrooted gene trees. We note that
previous work on invariants for statistical models in phylogenetics (Allman and Rhodes
2003; Cavender and Felsenstein 1987; Lake 1987) has focused on polynomial constraints
for site pattern probabilities; that is, probabilities that leaves of a gene tree display
various states (e.g., one of four states for DNA nucleotides) under models of charac-
ter change, given the topology and branch lengths of the gene tree. These approaches
have been particularly useful in determining identifiability of (gene) trees given se-
quence data under different models of mutation (Allman and Rhodes 2006; 2008; 2009;
Allman et al 2010a;b).
In this paper, our methods focus on understanding linear invariants and inequalities
for distributions of unrooted gene tree topologies under the multispecies coalescent
model. Here gene trees are branching patterns representing ancestry and descent for
genetic lineages, and are independent of mutations that may have arisen on these
lineages. This is therefore a novel application of invariants in phylogenetics.
2 Notation
Let X denote a set of |X| = n taxa, and let ψ+denote a rooted, binary, topological
species tree whose n leaves are labeled by the elements of X. If ψ+is further endowed
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with a collection λ+of strictly positive edge lengths for the n − 2 internal edges, then
σ+= (ψ+,λ+) denotes a rooted, binary, leaf-labeled, metric species tree on X. Note
that edge lengths in the species tree do not represent evolutionary time directly, but
are in coalescent units, that is units of τ/Ne, where τ is the number of generations and
Ne is the effective population size, the effective number of gene copies in a population
(Degnan and Rosenberg 2009). As pendant edge lengths do not affect the probability
of observing any topological gene tree, rooted or unrooted, under the multispecies
coalescent model with one individual sampled from each taxon, they are not specified
in λ+. To specify a particular species tree σ+, we use a modified Newick notation which
omits pendant edge lengths. For instance, a particular 4-taxon balanced metric species
tree is σ+= ((a,b):0.1,(c,d):0.05). Rooted 4- and 5-taxon species trees with branch
lengths which will be used later in this paper are depicted in Fig. 2. We refer to the
5-taxon tree shapes as balanced, caterpillar, and following Rosenberg, pseudocaterpillar.
((a,b):x,(c,d):y)
(a)
x
y
abcd
(((a,b):x,c):y,d)
(b)
x
y
abcd
(((a,b):x,c):y,(d,e):z)
balanced
x
y
z
(c)
abcd
e
((((a,b):x,c):y,d):z,e)
(d)
x
y
z
abcde
caterpillar
(((a,b):x,(d,e):y):z,c)
(e)
x
z
y
abdec
pseudocaterpillar
Fig. 2 Model species trees with branch lengths used to determine probabilities of unrooted
gene trees in this paper. The two 4-taxon species trees in (a) and (b) each have the same
unrooted topology, namely a tree with the ab|cd split. The three 5-taxon species trees in
(c)–(e) also share one unrooted topology, the topology with the splits ab|cde and abc|de.
Replacing ‘+’ with ‘−’ denotes suppressing the root, so that ψ−is the unrooted
binary topological species tree, λ−the induced collection of n−3 internal edge lengths
on ψ−, and σ−= (ψ−,λ−) is the unrooted metric species tree. An unrooted topology
can be specified by its nontrivial splits — the partitions of the taxa induced by removing
an internal edge of the unrooted tree. For example, T15in Fig. 1h has splits BE|ACD
and ABE|CD. A set of all taxa descended from a node in a rooted tree forms a clade,
the rooted analog of a split. For example, the rooted gene tree in Fig. 1a has 2-clades
{B,E} and {C,D} and the 3-clade {A,B,E}.
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For any set of taxa S ⊆ X, we let TSdenote the collection of all unrooted, binary,
leaf-labeled topological gene trees for the taxa S. We use the convention that while
lower-case letters denote taxa on a species tree, the corresponding upper-case letters
are used as leaf labels on a gene tree; Thus A denotes a gene from taxon a, etc. For
example, if X = {a,b,c,d}, then
TX= {AB|CD, AC|BD, AD|BC}.
Given any sort of tree (species/gene, rooted/unrooted, topological/metric) on X,
appending ‘(S)’ denotes the induced tree on the taxa S ⊆ X. By ‘induced tree’ here
we mean the tree obtained by taking the minimal subtree with leaves in S and then
suppressing all non-root nodes of degree 2. Instances of this notation include σ+(S),
σ−(S), ψ+(S), ψ−(S), and T(S).
3 The multispecies coalescent model
Several papers have given examples of applying the coalescent process to multiple
species or populations to derive examples of probabilities of rooted gene tree topologies
given species trees (Nei 1987; Pamilo and Nei 1988; Rosenberg 2002) with the general
case (for any n-taxon, rooted, binary species tree) given in (Degnan and Salter 2005).
We present the model here with only one individual sampled per taxon, as that will
be sufficient for our analysis.
Under the multispecies coalescent model, waiting times (going backwards in time)
until coalescent events (nodes in a rooted gene tree) are exponential random variables.
The rate for these variables is
?i
by enumerating all possible specifications of branches in which each coalescent event
occurs, and computing the probability of these events in each branch, treating each
branch as a separate population. In particular, the probability that i lineages coalesce
into j lineages within time t is represented by the function gij(t) (Tavar´ e 1984), which
is a linear combination of exponential functions:
2
?, with i the number of lineages “entering” a pop-
ulation, i.e., a branch on the species tree. Gene tree probabilities can be computed
gij(t) =
i
?
k=j
exp
?
−
?
k
2
?
t
?
(2k − 1)(−1)k−j
j!(k − j)!(j + k − 1)
k−1
?
m=0
(j + m)(i − m)
i + m
,1 ≤ j ≤ i.
(2)
Here t > 0 is time measured in coalescent units. The functions gij have the prop-
erty that for any i > 1 and any t > 0, gij(t), j = 1,...,i, is a discrete probabil-
ity distribution, that for any i > 1, limt→∞gi1(t) = 1, and that limt→0gii(t) = 1.
These last two properties express the ideas that given enough time, all lineages even-
tually coalesce (there is only one lineage remaining in a population) and that over very
short time intervals, it is very likely that no coalescent events occur. Finally, note that
gii(t) = exp(−i(i − 1)t/2).
As an example of using this function to determine rooted gene tree probabilities,
consider the rooted caterpillar species tree ((((a,b):x,c):y,d):z,e) of Fig. 2d, and the
rooted gene tree ((((B,E),A),C),D). Since this gene tree requires a specific ordering
of coalescences, and the first of these can only occur in the population above the
root of the species tree, the only scenario to consider is that shown in Fig. 1c. In
the population ancestral to species a and b, there are two lineages which must fail
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to coalesce in time x, and this event has probability g22(x) = exp(−x). Similarly,
the events in the populations with durations y and z have probabilities exp(−3y)
and exp(−6z), respectively, because no lineages coalesce in those intervals. For the
population ancestral to the root, all lineages eventually coalesce, and the probability
for events in this population is the probability of observing the particular sequence of
coalescence events, which is??5
to work with transformed branch lengths, where if a branch has length x, we set
X = exp(−x). Using this notation, the rooted gene tree has probability XY3Z6/180.
As another example, consider the gene tree (((B,E),A),(C,D)) given the same
species tree, ((((a,b):x,c):y,d):z,e). For this rooted gene tree to be realized, either C
and D coalesce as depicted in Fig. 1a, in the population immediately below the root
(which we call the “near the root” population), or C and D coalesce above the root.
Regardless, all other coalescent events must occur in the population above the root. We
therefore divide the calculation of the rooted gene tree topology into these two cases. If
all coalescent events occur above the root, the rooted gene tree probability is calculated
as in the preceding paragraph, except that there are three possible orders in which the
coalescent events could occur to realize the rooted gene tree, and the probability for
this case is thus XY3Z6/60. In the case where C and D coalesce “near the root,” there
are no coalescent events in the populations with lengths x and y, thus contributing a
factor of exp(−x − 3y) to the probability. The probability for events near the root is
?4
entering the population above the root of the species tree, the one sequence of coalescent
events that results in the gene tree topology has probability??4
tree (((a,b):x,c):y,d):z,e) is therefore
2
??4
2
??3
2
??2
2
??−1= 1/180. The probability of the rooted
gene tree given the species tree is therefore exp(−x−3y−6z)/180. It is often convenient
2
?−1g43(z), where the coefficient is the probability that of the four lineages entering
the population, the two that coalesce are C and D. Because there are four lineages
2
??3
2
??2
2
??−1= 1/18. The
total probability of the rooted gene tree topology (((B,E),A),(C,D)) given the species
g22(x)g33(y)1
?4
2
?g43(z)
1
?4
2
??3
2
??2
2
? + g22(x)g33(y)g44(z)
3
??3
?5
2
??4
2
2
??2
2
?
=XY31
6(2Z3− 2Z6)1
54XY3Z3−
18+1 60XY3Z6
=1
1
540XY3Z6.
Probabilities of the other rooted gene trees in Fig. 1 can be worked out similarly by
considering a small number of cases for each tree. Methods for enumerating all possible
cases have been developed using the concept of coalescent history, a list of populations
in which the coalescent events occur (Degnan and Salter 2005). Each coalescent history
h has a probability of the form
c(h)
n−2
?
b=1
gi(h,b),j(h,b)(xb) (3)
where xbis the length of internal edge b of the species tree and c(h) is a constant that
depends on the coalescent history h and the topologies of the gene and species trees,
but does not depend on the branch lengths xb. This expression is a linear combination
of products of terms exp[−k(k−1)xb/2], k = 2,...,n−1, so using the transformations
Xb= exp(−xb), probabilities of coalescent histories can thus be written as polynomials
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in the transformed branch lengths of the species tree. Because gene tree probabilities
are sums of probabilities of coalescent histories, gene tree probabilities can also be
written as polynomials in the transformed branch lengths.
Finally, unrooted gene tree probabilities, which are sums of rooted gene tree prob-
abilities, can also be expressed as polynomials in the transformed branch lengths. We
thus can derive polynomial expressions for the probabilities of unrooted gene trees
given a species tree.
4 Results
The unrooted topological gene tree distribution under the multispecies coalescent
model on species tree σ+, with one lineage sampled per species, will be denoted by
P = Pσ+, so that P(T) denotes the probability of observing gene tree T ∈ TX.
For ease of exposition, we assume throughout this section that the species tree σ+
is binary. See Section 5 for the polytomous case.
4.1 4-taxon trees
We first consider the case of four taxa, and so let X = {a,b,c,d}. Using non-trivial
splits as indices, the set of gene trees is
TX= {TAB|CD,TAC|BD,TAD|BC}.
With four taxa, there are only two shapes for species trees: the balanced tree, with
two clades of size 2 (Fig. 2a); and the rooted caterpillar tree with a 2-clade nested inside
a 3-clade (Fig. 2b). Of the 15 possibilities for ψ+, there are three labeled balanced tree
topologies, and 12 labeled caterpillar topologies. It is only necessary to compute gene
tree probabilities for a single labeling of the leaves of each species tree shape, since
permuting labels immediately gives the distribution for other choices.
For a balanced tree σ+= (((a,b):x,(c,d):y) shown in Fig. 2a, one computes, as
described in the previous section, that the gene tree distribution is given by
Pσ+(TAB|CD) = 1 −2
3e−(x+y),
Pσ+(TAC|BD) = Pσ+(TAD|BC) =1
3e−(x+y).
For a rooted caterpillar species tree σ+= (((a,b):x,c):y,d) shown in Fig. 2b, one
finds
Pσ+(TAB|CD) = 1 −2
3e−x,
Pσ+(TAC|BD) = Pσ+(TAD|BC) =1
3e−x.
Thus for any 4-taxon species tree, from the distribution Pσ+ one can identify the
unrooted species tree topology ψ−as that of the most probable unrooted gene tree T.
The one internal edge length on ψ−(i.e., x+y in the balanced case, x for the caterpillar)
can be recovered as −log?3
2(1 − P(T))?. Thus σ−= (ψ−,λ−) is identifiable.
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Furthermore σ+is not identifiable since the above calculations show that for any
x > 0, yi> 0, and x > z > 0 the following rooted species trees produce exactly the
same unrooted gene tree distribution:
((a,b):x,c):y1,d),
((a,b):x,d):y2,c),
((c,d):x,a):y3,b),
((c,d):x,b):y4,a),
((a,b):z,(c,d):x − z).
We summarize this by:
Proposition 3 For |X| = 4 taxa, σ−is identifiable from Pσ+, but σ+is not.
We note that if the unrooted gene trees are ultrametric with known branch lengths,
then their rooted topologies are known by midpoint rooting (Kim et al 1993), and thus
σ+is identifiable from unrooted ultrametric 4-taxon gene trees.
4.2 Linear invariants and inequalities for unrooted gene tree probabilities for 5-taxon
species trees
To establish identifiability of all parameters when there are at least 5 taxa, we will
argue from the 5-taxon case. In this base case we will use an understanding of linear
relationships — both equalities and inequalities — that hold between gene tree prob-
abilities. The relationships that hold for a particular gene tree distribution reflect the
species tree on which it arose.
In this section, we determine all linear equations in gene tree probabilities for each
of the three shapes of 5-leaf species trees. Following phylogenetic terminology, these are
the linear invariants of the gene tree distribution. We emphasize that these invariants
depend only on the rooted topology, ψ+, of the species tree, and not on the branch
lengths λ+. Although some of these invariants arise from symmetries of the species
tree, others are less obvious. Nonetheless, we give simple arguments for all, and show
that there are no others. In addition, we provide all pairwise inequalities of the form
ui> ujfor the three model species trees in Figs. 2c–e.
With X = {a,b,c,d,e}, there are 15 unrooted gene trees in TX, which we enumerate
in Table 5 of Appendix A. Probabilities for each of the 15 unrooted gene trees are
obtained by summing probabilities of seven of the 105 rooted 5-taxon gene trees, as
shown in Tables 4 and 5 of Appendix A. In Appendix B formulas for the unrooted gene
tree distribution are given for one choice of a leaf-labeling of each of the three possible
rooted species tree shapes. Noticing that many of the gene tree probabilities are equal,
one might hope that which ones are equal would be useful in identifying the species
tree from the distribution.
For each species tree one can computationally, but entirely rigorously, determine a
basis for the vector space of all linear invariants. We report such a basis for each of the
species tree shapes below, in Tables 1-3. Only for one of the tree shapes is an additional
invariant that is not immediately noticeable produced by this calculation. While our
computations were performed using the algebra software Singular (Greuel et al 2009),
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many other packages would work as well, or one could do the calculations without
machine aid.
In the tables and discussion below, we omit mention of the trivial invariant,
15
?
i=1
Pσ+(Ti) = 1,
which holds for any choice of σ+. We instead only give a basis for the homogeneous
linear invariants.
We use the following observation.
Lemma 4 If all coalescent events occur above the root (temporally before the MRCA
of all species) of a 5-taxon species tree, then all 15 of the unrooted topological gene trees
are equally likely.
Proof If all coalescent events occur above the root, then regardless of the species tree,
we are considering five labeled lineages entering the ancestral population, and then
coalescing. Because all unrooted gene trees have the same unlabeled shape, all coales-
cent histories leading to one gene tree correspond to equally likely coalescent histories
producing another, by simply relabeling lineages.⊓ ⊔
Note that the claim of this lemma is special to five taxa. For six taxa, with two different
unrooted gene tree shapes possible, the analogous statement is not true.
4.2.1 Balanced species tree
Suppose ψ+= (((a,b),c),(d,e)), as depicted in Fig. 2c. Because σ+is invariant under
interchanging taxa a and b, any two gene trees that differ by transposing leaves A and
B must have the same probability. Similarly, interchanging D and E on a gene tree
cannot change its probability. We refer to the first permutation of labels using cycle
notation as (ab), and the second as (de). More formally, assuming generic values for
λ+, the symmetry group of σ+is the 4-element group generated by the transpositions
(ab) and (de), and the gene tree probability distribution must be invariant under the
action of this group on gene trees. These symmetries thus give ‘explanations’ for many
invariants holding.
A different explanation for some invariants is that some unrooted gene trees can
only be realized if all coalescent events occur above (more anciently than) the root of
the species tree. For example, any realization of the gene tree T15with splits BE|ACD
and ABE|CD (Fig. 1h) requires that the first (most recent) coalescent event either
joins lineages B and E, or joins C and D. Because both of these events can only occur
above the root, all events must take place above the root. Another such gene tree is T11,
with splits AE|BCD and ACE|BD. Thus by Lemma 4 the unrooted gene trees T11
and T15must have the same probability, even though they do not differ by a symmetry
as described in the last paragraph. We refer to this reasoning as the “above the root”
argument.
Some invariants can be explained in several ways. For example, the same invariant
might be explained by two different symmetries or by both a symmetry and an above-
the-root argument. In Table 1, we list a basis for homogeneous linear invariants, and
give only one explanation for each. Here ui= P(Ti).
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12
Table 1 Invariants for the rooted species tree ψ+= (((a, b),c),(d,e))
Invariant
u14− u15= 0
u11− u15= 0
u10− u15= 0
u9− u12= 0
u8− u15= 0
u7− u15= 0
u6− u12= 0
u5− u12= 0
u4− u13= 0
u2− u3= 0
Explanation
(de)
above root
(ab)
(de)
above root
(ab)(de)
(ab)(de)
(ab)
(ab)
(de)
These equalities give the following equivalence classes of unrooted gene trees ac-
cording to their probabilities:
{T1},{T2,T3},{T4,T13},{T5,T6,T9,T12},{T7,T8,T10,T11,T14,T15}.
For any branch lengths on this species tree, we also observe the inequalities
u1> u2,u4> u5> u7. (4)
These inequalities were found by first expressing the probability of each Tias a
sum of positive terms corresponding to coalescent histories, such as expression (3), and
then, by comparing coefficients in these sums, determining instances in which ui> uj
must hold. Intuitively, this means that any realization of Tjcorresponds to a realization
of Ti, but that there are additional ways that Tican be realized.
The inequalities in (4) can all be checked by elementary arguments using the explicit
formulas of Appendix B. For instance, since X,Y,Z ∈ (0,1),
u1− u2= 1 −2
3X − Y Z +1
> 1 − Y Z −1
2XY Z +1
6XY3Z = 1 − Y Z −1
3−1
6X(4 − 3Y Z − Y3Z)
6(4 − 3Y Z − Y3Z) =1
6Y (3 − Y2) > 0.
6Y Z(3 − Y2)
>1
3−1
In particular, there is always a 6-element equivalence class of trees which has the
strictly smallest probability associated with it, and a 4-element class which has the
next smallest probability associated to it. While the class associated to the largest
probability is always a singleton, these inequalities do allow for the remaining two
classes of size 2 to degenerate to a single class of size 4.
Numerical examples can be used to show that there are no inequalities of the form
ui> ujthat hold for all branch lengths X, Y , and Z that are not listed in (4).
4.2.2 Caterpillar species tree
Suppose ψ+= ((((a,b),c),d),e), as depicted in Fig. 2d. Then the symmetry group of
the tree is generated by (ab), and has only two elements.
Although no unrooted gene trees require that all coalescent events occur above
the root of this species tree, there are gene trees that require that all events be either
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13
above the root or “near the root” in the following sense. Consider the gene tree T15
with splits BE|ACD and ABE|CD (Fig. 1h). This gene tree can be realized either by
all events occurring above the root (in which case either the BE coalescence or the CD
coalescence could be first), or by 1, 2, or 3 events occurring in a specific order in the
near-the-root population which is ancestral to species a,b,c, and d but not to e, with
all further events above the root. For example, if there are two coalescent events in this
population, then the gene tree must have ((CD)A) as a subtree (Fig. 1b,e,f), and C
and D must coalesce most recently followed by the coalescence of A. In case 1, 2, or 3
events do occur below the root, these must be in the specific order 1) CD coalesce, 2)
ACD coalesce, 3) ABCD coalesce. Another gene tree which leads to a similar analysis
of how coalescent events must occur for the gene tree to be realized is T14, with splits
BD|ACE, ABD|CE. Consequently, T14has the same probability as T15, even though
these two gene trees do not differ by a symmetry. Similar arguments apply to trees T7,
T8, T10, and T11. The near-the-root argument and symmetry between a and b explain
all linear invariants but the last in Table 2.
Table 2 Invariants for the rooted species tree ψ+= ((((a, b),c),d),e)
Invariant
u14− u15= 0
u11− u15= 0
u10− u15= 0
u8− u15= 0
u7− u15= 0
u6− u9= 0
u5− u12= 0
u4− u13= 0
u2− u3+ u9− u12= 0
Explanation
near root
near root
(ab)
near root
near root
(ab)
(ab)
(ab)
marginalization
To explain the last invariant in Table 2, we provide a marginalization argument.
We use the fact that for 4-taxon trees the two unrooted gene trees that are inconsistent
with the species tree are equiprobable. Thus, marginalizing over a to trees on {b,c,d,e},
we have that P(TBD|CE) = P(TBE|CD). Hence,
u2+ u6+ u7+ u11+ u14= u3+ u5+ u8+ u10+ u15.
Because the last 3 terms on each side are equal to u15, we may cancel those. Replacing
u6with u9, and u5with u12, then gives the last invariant in the table.
Table 2 yields the following equivalence classes of gene trees according to their
probabilities:
{T1},{T2},{T3},{T4,T13},{T5,T12},{T6,T9},{T7,T8,T10,T11,T14,T15}.
We also observe that the inequalities
u1> u2,u4> u5> u7,
u3> u2,u6> u5> u7
(5)
hold for all branch lengths on this species tree, and that there are no other inequalties
of the form ui> ujthat hold for all branch lengths, by arguments similar to those for
the balanced tree.
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4.2.3 Pseudocaterpillar species tree
Suppose ψ+= (((a,b),(d,e)),c), as depicted in Fig. 2e. Then the symmetry group of
the tree σ+is generated by (ab) and (de), and has four elements. (Note that inter-
changing the two cherries, for instance by (ad)(be), is a symmetry of ψ+, but is not a
symmetry of σ+for generic edge lengths.)
While no unrooted gene trees require that all coalescent events occur above the
root of this species tree, some unrooted gene trees require that all events be either near
the root or above the root. The gene tree T15, with splits BE|ACD and ABE|CD, can
be realized either by all events occurring above the root (in which case either the BE
coalescence or the CD coalescence could be first), or by 1, 2, or 3 events occurring in
a specific order in the population ancestral to species a,b,c, and d but not to e, with
all further events occurring above the root. In case 1, 2, or 3 events do occur below the
root, these must be in the specific order 1) BE coalesce, 2) ABE coalesce, 3) ABDE
coalesce. Another gene tree which leads to a similar analysis of how coalescent events
must occur for the gene tree to be realized is T12, with splits AE|BCD, ADE|BC.
Thus T12and T15are equiprobable, even though they do not differ by a symmetry.
A basis for homogeneous linear invariants of unrooted gene tree probabilities, along
with explanations for each is given in Table 3.
Table 3 Invariants for the rooted species tree ψ+= (((a, b),(d,e)),c)
Invariant
u14− u15= 0
u12− u15= 0
u10− u15= 0
u9− u15= 0
u8− u11= 0
u7− u15= 0
u6− u15= 0
u5− u15= 0
u4− u13= 0
u2− u3= 0
Explanation
(de)
near root
(ab)
near root
(ab)
(ab)(de)
near root
near root
(ab)
(de)
We thus obtain the following equivalence classes of unrooted gene trees according
to their probabilities:
{T1},{T2,T3},{T4,T13},{T8,T11},{T5,T6,T7,T9,T10,T12,T14,T15}.
For all branch lengths on this species tree, we also observe the inequalities
u1> u2,u4,u8> u5
(6)
and note that there are no other inequalities of the form ui> uj that hold for all
possible branch lengths. In particular, the 8-element equivalence class of trees always
has the strictly smallest probability associated with it.
4.3 Species Tree Identifiability for 5 or more Taxa
We will use several times the following observation, which is clear from the structure
of the coalescent model. (In fact, this has already been used in Section 4.2.2 in the
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marginalization argument explaining a linear invariant for the caterpillar tree.) While
we state the lemma for unrooted gene trees, there is of course a similar statement for
the distribution of rooted gene trees.
Lemma 5 If S ⊆ X and T′∈ TS, then
Pσ+(S)(T′) =
?
T∈TX
T(S)=T′
Pσ+(T).
As a consequence of the analysis for 4-taxon trees in Section 4.1, we obtain the
following.
Corollary 6 For any X, Pσ+ determines σ−.
Proof We assume |X| ≥ 4, since otherwise there is nothing to prove. For any quartet
Q ⊆ X of four distinct taxa, by Lemma 5, Pσ+ determines Pσ+(Q). Thus σ−(Q)
is determined by Proposition 3. Thus all unrooted quartet trees induced by ψ−are
determined, along with their internal edge lengths. That all induced quartet topologies
determine the topology ψ−is well known (Steel 1992). Because each internal edge of
ψ−is the internal edge for some induced quartet tree, λ−is determined as well.⊓ ⊔
For the remaining arguments to determine σ+, we may assume that σ−is already
known. We focus first on the |X| = 5 case, and thus assume that X = {a,b,c,d,e} and
that ψ−has non-trivial splits ab|cde and abc|de.
Proposition 7 For |X| = 5 the rooted species tree topology ψ+is determined by Pσ+.
Proof From Section 4.2, for generic values of λ+, the caterpillar leads to seven distinct
gene tree probabilities, with class sizes 1,1,1,2,2,2,6; the pseudocaterpillar gives five
distinct probabilities, with class sizes 1,2,2,2,8; and the balanced tree gives five dis-
tinct probabilities, with class sizes 1,2,2,4,6. Thus the (unlabeled) shape of ψ+can be
distinguished for generic edge lengths. However, for certain values of these parameters
the classes can degenerate, by merging.
To see that the tree shapes can be distinguished for all parameter values, observe
that the inequalities (4)–(6) of Section 4.2 on gene tree probabilities ensures the class
associated to the smallest probability always has size 8 for the pseudocaterpillar, while
for the other shapes the size of this class is always 6. Moreover, for the caterpillar
and balanced trees the size of the class associated to the second smallest probability
must be exactly 2 and 4, respectively. Thus, these class sizes allow us to determine the
unlabeled, rooted shape (balanced, caterpillar, or pseudocaterpillar) of the species tree.
In addition, from Corollary 6, we also know the labeled, unrooted topology (i.e., the
splits) of the species tree, ψ−. To determine the labeled, rooted topology, we consider
cases depending on the unlabeled, rooted shape determined from the class sizes.
If the species tree is balanced, from the splits we know that ψ+= (((a,b),c),(d,e))
or ψ+= ((a,b),(c,(d,e))). But the gene tree T7, with splits AD|BCE and ABD|CE,
can be realized on the first of these species trees only if all coalescent events occur
above the root; on the second species tree, T7can be realized other ways as well. Thus
T7would fall into the 6-element class of least probable gene trees for the first but not
the second species tree. This then determines ψ+.
For a caterpillar species tree, from the splits we know ψ+has as its unique 2-
clade either {a,b} or {d,e}. By considering the cherries on the two gene trees in the
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class of those with the second smallest probability, we see the 2-clade is determined as
those taxa that appear in cherries with c. For simplicity, we henceforth suppose that
the 2-clade is found to be {a,b}. Thus, ψ+= ((((a,b),c),d),e) or ((((a,b),c),e),d).
Then from the inequality (5) in Section 4.2.2, we find that P(T3) > P(T2) if ψ+=
((((a,b),c),d),e), while this inequality is reversed if ψ+= ((((a,b),c),e),d).
In the case of the pseudocaterpillar species tree, because the splits of ψ−are known,
there is only one possibility for ψ+. Thus ψ+is determined.⊓ ⊔
Proposition 8 For |X| = 5, Pσ+ determines σ+= (ψ+,λ+).
Proof By Proposition 7 , ψ+is determined. From Corollary 6, λ−is also determined.
Thus all elements of λ+except for the edges incident to the root are determined. In
the balanced case, the sum of these two unknown edge lengths is determined, but
in the other cases we have yet to determine any information about the single such
non-pendant edge length. We therefore consider each of these cases in order.
If ψ+is balanced, we may assume σ+= (((a,b):x,c):y),(d,e):z), with y,z still to
be determined. As the unrooted internal edge length y + z is known, it is enough to
determine y. From the gene tree probabilities in Appendix B.1, it follows that
XY Z = 6u5+ 9u7,
XY3Z = 15u7.
Thus,
y = −log(Y ) =1
2log
?2u5+ 3u7
5u7
?
.(7)
If ψ+is a rooted caterpillar, we may assume σ+= ((((a,b):x,c):y,d):z,e). Only z
remains to be determined. Using the explicit formulas for gene tree probabilities given
in Appendix B.2, one checks that
XY3= 3(−u2+ u3+ 5u7)
XY3Z6= 15(u2− u3+ u7)
and thus
z = −log(Z) =1
6log
?−u2+ u3+ 5u7
5u2− 5u3+ 5u7
?
. (8)
If ψ+is the pseudocaterpillar, we may assume σ+= (((a,b):x,(d,e):y):z,c), with
z still to be determined. The gene tree probabilities listed in Appendix B.3 show that
XY = 12u5+ 3u8
XY Z6= 30u5− 15u8.
Thus,
z = −log(Z) =1
6log
?
4u5+ u8
10u5− 5u8
?
.(9)
Note that equations (11)–(13) of Appendix B can be used to show that the arguments
of the logarithms in equations (7)–(9) are always strictly greater than 1.⊓ ⊔
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While this proof used particular formulas to identify the remaining edge lengths in
λ+, note that many variants could have been used in their place. This simply reflects
the many algebraic relationships (both linear and non-linear invariants) between the
various gene tree probabilities.
With the |X| = 5 case completed, we obtain the general result.
Theorem 9 The unrooted topological gene tree distribution Pσ+ arising from the mult-
species coalescent model for samples of one lineage per taxon determines the metric
species tree σ+provided |X| ≥ 5. If |X| = 4, Pσ+ determines only the unrooted metric
species tree σ−.
Proof By Corollary 6, σ−= (ψ−,λ−) is determined.
If |X| ≥ 5, consider a specific edge e of ψ−, and all 5-taxon subsets S ⊆ X such
that the induced unrooted tree ψ−(S) has e as an edge. If the root, ρ, of ψ+lies on
e, then the root of ψ+(S) is also ρ and thus the root of ψ+(S) lies on e for all such
S. If ρ does not lie on e, then there exists an S with the root of ψ+(S) not on e. To
see this, for any set Q ⊂ X of four taxa which distinguishes e (Steel 1992, Proposition
6), choose x ∈ X \ Q so that the MRCA of S = Q ∪ {x} is ρ. Then ψ+(S) has root ρ,
which is not on e.
Thus using Lemma 5 and Proposition 8 to determine the root location of such
ψ+(S) for each e, we can determine ψ+. Then the length of any internal edges incident
to the root of ψ+can be recovered by choosing a 5-taxon subset S such that ψ+(S) has
these edges, and applying Lemma 5 and Proposition 8 again. Thus σ+is determined.
Proposition 3 gives the case |X| = 4.⊓ ⊔
Theorem 9 gives an alternate approach to establishing Corollary 2, in cases with
|X| ≥ 5, since the distribution of rooted gene trees determines that of unrooted gene
trees.
Theorem 9 can also be used to show that if multiple lineages are sampled from
some or all of the taxa, then the unrooted gene tree distribution contains additional
information on the species tree, as follows.
Corollary 10 Consider a species tree on taxon set X, and, for some ℓi > 0, the
distribution of unrooted topological gene trees under a multispecies coalescent model of
samples of ℓiindividuals from taxon i. Suppose that either |X| ≥ 4 and that there is at
least one i such that ℓi≥ 2, or that |X| = 3 and that there are at least two values of
i such that ℓi≥ 2. Then the gene tree distribution determines the species tree’s rooted
topology, internal edge lengths, and for any taxon with li> 1 the length of the pendant
edge leading to taxon i.
Proof We may assume all ℓiare either 1 or 2, by marginalizing over any additional
individuals sampled, if necessary.
Construct an extended species tree by attaching to any leaf i for which li= 2 a
pair of edges descending to new leaves labelled i1and i2, so the extended species tree
has ℓ =?n
The lengths of the new pendant edges in the extended tree can be chosen arbitrarily,
or left unspecified. Then a coalescent process on the extended ℓ-taxon tree with one
sample per leaf leads to exactly the same distribution of topological gene trees as the
the multiple-sample process on the original species tree.
Applying Theorem 9 to the extended tree, we obtain the result.
i=1ℓileaves. The pendant edge leading to taxon i in the original species
tree becomes an internal edge on the extended tree, and retains its previous length.
⊓ ⊔
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5 Nonbinary species trees
The results for binary species trees generalize to nonbinary species trees as well. When
species trees are allowed to be nonbinary, there are two unlabeled 3-taxon tree shapes,
five unlabeled 4-taxon tree shapes, and 12 unlabeled 5-taxon tree shapes (Cayley 1857).
Probabilities of binary, unrooted gene tree topologies given a nonbinary species tree
can be obtained by considering the limiting probability as one or more branch lengths
go to zero in the formulas derived for binary species trees. We note that under the
standard Kingman coalescent, gene trees, which depend on exponential waiting times,
are still binary with probability 1 even when the species tree has polytomies.
For the 3-taxon species tree, ((a,b):t,c), letting t → 0, the rooted gene tree proba-
bilities are each 1/3 in the limit. Thus the unresolved 3-taxon rooted species tree can be
identified from the gene tree distribution from the presence of three equal probabilities;
whereas for a resolved species tree, exactly one gene tree has probability greater than
1/3. Similarly, polytomies in any larger species tree can be identified by considering
rooted triplets. A species tree node has three or more descendants if the three rooted
gene trees obtained from sampling one gene from three distinct descendants of the node
have equal probabilities.
For 4-taxon species trees, the completely unresolved topology (a,b,c,d) can not
be distinguished from the partially unresolved ((a,b,c):y,d) from unrooted gene tree
probabilities as both result in equal probabilities of the three binary, unrooted gene
trees on these taxa. Similarly, the resolved species trees (((a,b):x,c):y,d) and the par-
tially unresolved ((a,b):x,c,d) yield the same unrooted gene tree probabilities, with
Pσ+(TAB|CD) = 1−2
case, that 4-taxon unrooted gene tree probabilities identify the unrooted (possibly un-
resolved) species tree, but do not identify the root. Thus Proposition 3 is still valid
when σ+is nonbinary.
Identifiability of possibly-nonbinary rooted species trees for 5 or more taxa from
probabilities of unrooted gene tree topologies can be established using arguments sim-
ilar to those of the binary case. While we defer the detailed proofs to Appendix C, we
state these results as follows:
3e−x. These observations lead to the conclusion, as in the binary
Proposition 11 Proposition 3, Corollary 6, Propositions 7 and 8, Theorem 9, and
Corollary 10 remain valid if σ+is nonbinary.
We note that a species tree with a polytomy is equivalent to a model of a resolved
species tree with one or more branch lengths set equal to zero, and therefore that a
resolved species tree and a polytomous species tree can be regarded as nested models.
Although it might be difficult to distinguish polytomous versus resolved species trees
from finite amounts of data, the nested relationship of these models suggests that
likelihood ratios could be used to determine whether an estimated species tree branch
length is significantly greater than 0. A previous study (Poe and Chubb 2004) argued
for the hypothesis of a species-level polytomy in early bird evolution by using likelihood
ratios to test whether gene trees had branch lengths significantly greater than 0 at
multiple loci. Since gene trees are theoretically expected to be resolved under the
coalescent model, an alternative procedure would be to use probabilities of gene trees
under the polytomous and resolved species trees and perform a likelihood ratio test for
whether an estimated species tree branch length is significantly greater than 0.
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6 Discussion
Under standard models of sequence evolution, the distribution of site patterns of DNA
does not depend on the position of the root of the gene tree on which the sequence
evolve (this is sometimes called the “pulley principle” (Felsenstein 1981)). Inference of
the root of a gene tree requires additional assumptions, such as that of a molecular clock
(mutation occurs at a constant rate throughout the tree), or inclusion of an outgroup
taxon in the analysis, so that the root may be assumed to lie where the outgroup joins
all other taxa in the study. We have shown, however, that under the coalescent model
with five or more species, the distribution of unrooted topological gene trees preserves
information about both the rooted species tree and its internal branch lengths. Thus
in multilocus studies in which many gene trees are inferred, it is theoretically possible
to infer the rooted metric species tree even in the absence of a molecular clock, known
outgroups, or any metric information on the gene trees. While for some data sets it can
be difficult to obtain either reliable roots or evolutionary times for branches of gene
trees, these issues are not fundamental barriers to species tree inference.
Although we have shown the theoretical possibility of identifying rooted species
trees from unrooted gene trees by using linear invariants, we emphasize that we do
not propose using these invariants as a basis for inference. Invariants of gene tree
distributions are functions of their exact probabilities under the model — from finite
data sets, gene trees are inferred with some error, and empirical estimates of gene
tree probabilities from a finite number of gene trees might not satisfy invariants or
inequalities that apply to the exact distribution. Moreover, many non-linear invariants
which are not discussed in this paper (and not yet fully understood) further constrain
the form of the gene tree distribution.
In practice, very large numbers of loci might be needed to obtain approximate esti-
mates of gene tree probabilities, and there must be considerable gene tree discordance
in order to estimate probabilities of less probable unrooted gene trees. For example,
in an often-analyzed 106-gene yeast dataset (Rokas et al 2003), analyzing only the five
species about which there is the most conflict, (S. cerevisiae, S. paradoxus, S. mikatae,
S. kudriavzevil, and S. bayanus), yields the same unrooted gene tree for all 106 loci when
inferred using maximum likelihood under the GTR+Γ +I model without a molecular
clock. If all observed gene trees have the same unrooted topology, then there is not
enough information to infer the rooted species tree. Other data have shown more con-
flict in unrooted gene trees, such as a 162-gene dataset for rice (Cranston et al 2009),
in which 99 of 105 rooted 5-taxon gene trees were represented in the Bayesian 95%
highest posterior density (HDP) set of trees.
If species tree branches are too long, it will not be possible to recover the rooted
species tree from finite data. For example, if the species tree is (((a,b):x,c):y,d):z,e),
where y is sufficiently large, every observed gene tree (for a finite number of loci) might
have the ABC|DE split. Being able to determine that e is the outgroup would require
observing conflicting splits, such as that ABD|CE is more probable than ABE|CD.
However, if y is large, these conflicting splits are likely to never be observed, making
it difficult to distinguish between rooted topologies (((a,b),c),d),e), (((a,b),c),e),d),
and (((a,b),c),(d,e)).
On the other hand, if branches are too short, it might be difficult to distin-
guish between certain rooted species trees, such as between (((a,b),c),(d,e)) and
((a,b),(c,(d,e))) when the node immediately ancestral to c is very close to the root.
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20
Further study would be needed for a precise understanding of how extreme branch
lengths affect the number of gene trees needed for reliable inference of the species tree.
We note, however, that even when the rooted species tree cannot be fully inferred with
great certainty, some rooted aspects of the tree might be recoverable. For example in
the case of the rooted trees above, one might infer that (a,b) and (d,e) are rooted
cherries on the tree, even if the placement of taxon c with respect to the root remains
unknown.
We again emphasize that invariants are not the most promising approach for infer-
ring species trees from finite data, and that a maximum likelihood (ML) or Bayesian
approach might be more appropriate. Given a set of sufficiently conflicting unrooted
topological gene trees inferred by standard methods and then assumed to be correct,
the rooted species tree could be inferred using ML, where the likelihood of the species
tree is
(2n−5)!!
L(σ+) ∝
?
i=1
uni
i
(10)
where there are n taxa and the ith unrooted gene tree topology is observed nitimes
with?
However, this 2-stage approach of gene tree inference followed by species tree inference
does not take into account uncertainty in the gene trees, or cases in which inferred
gene trees are not fully resolved. If there is not enough information in the sequences
to estimate resolved gene trees, an approximation to equation (10) would be to either
randomly resolve the tree if there are very many loci (as is often done in software
implementing quartet puzzling (Strimmer and von Haeseler 1996) or neighbor joining
(Saitou and Nei 1987)); or, if an unresolved gene tree has k resolutions, let the locus
contribute a count of 1/k to each resolution.
To better utilize the information in the unrooted gene trees, an attractive, but
computationally more intensive, approach would use a Bayesian framework in which
the posterior distribution of the rooted species tree is determined from posterior distri-
butions of gene trees, thus taking into account uncertainty in the estimated gene trees.
Cases in which ML would return an unresolved gene tree would likely correspond to a
posterior distribution of gene trees with substantial support on more than one topol-
ogy. Thus, instead of each locus contributing a count of one gene tree topology, it
contributes fractional proportions to several topologies. In cases in which the gene tree
distributions carry little information about the root of the species tree, the posterior
distribution of the species tree would indicate this uncertainty by spreading the pos-
terior mass over several species trees. The results of the present paper suggest that
it is possible to extend current model-based methods of inferring rooted species trees
(e.g., BEST (Liu and Pearl 2007) and STEM (Kubatko et al 2009)) to cases where
only unrooted gene trees can be estimated.
Finally we note that invariants have a potential use in testing the fit of the multi-
species coalescent model to a dataset. As noted in (Slatkin and Pollack 2008), processes
such as population subdivision can lead to asymmetry in the probabilities of the two
nonmatching rooted gene trees in the case of three taxa, thus violating the invariant
in equation (1). As shown in this paper, similar invariants can be obtained for larger
number of species even when only unrooted gene trees are available, thus allowing the
testing of the fit of the multispecies coalescent model in situations more general than
the rooted 3-taxon setting.
ini = N the total number of loci. The probability uiof the ith gene tree
depends on the species tree topology and branch lengths as outlined in Section 3.
Page 21
21
Acknowledgements The authors thank the Statistical and Applied Mathematical Sciences
Institute, where this work was begun during its 2008-09 program on Algebraic Methods in
Systems Biology and Statistics. We also thank two anonymous reviewers, one of whom sug-
gested the extension to nonbinary trees. ESA and JAR were supported by funds from the
National Science Foundation, grant DMS 0714830, and JAR by an Erskine Fellowship from
the University of Canterbury. JHD was funded by the New Zealand Marsden Fund. All authors
contributed equally to this work.
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A Tables for 5-taxon trees
Table 4 The 105 rooted gene trees on 5 species.
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
R16
R17
R18
R19
R20
R21
R22
R23
R24
R25
R26
R27
R28
R29
R30
R31
R32
R33
R34
R35
((((A, B),C),D),E)
((((A, B),C),E),D)
((((A, B),D),C),E)
((((A, B),D),E),C)
((((A, B),E),C),D)
((((A, B),E),D),C)
((((A, C),B), D),E)
((((A, C),B), E),D)
((((A, C),D),B),E)
((((A, C),D),E),B)
((((A, C),E),B),D)
((((A, C),E),D),B)
((((A, D),B), C),E)
((((A, D),B), E),C)
((((A, D),C),B),E)
((((A, D),C),E),B)
((((A, D),E),B),C)
((((A, D),E),C),B)
((((A, E),B),C),D)
((((A, E),B),D),C)
((((A, E),C),B),D)
((((A, E),C),D),B)
((((A, E),D),B),C)
((((A, E),D),C),B)
((((B, C),A),D),E)
((((B, C),A),E),D)
((((B, C),D),A),E)
((((B, C),D),E),A)
((((B, C),E),A),D)
((((B, C),E),D),A)
((((B, D),A),C),E)
((((B, D),A),E),C)
((((B, D),C),A),E)
((((B, D),C),E),A)
((((B, D),E),A),C)
R36
R37
R38
R39
R40
R41
R42
R43
R44
R45
R46
R47
R48
R49
R50
R51
R52
R53
R54
R55
R56
R57
R58
R59
R60
R61
R62
R63
R64
R65
R66
R67
R68
R69
R70
((((B, D),E),C),A)
((((B, E),A),C),D)
((((B, E),A),D),C)
((((B, E),C),A),D)
((((B, E),C),D),A)
((((B, E),D),A),C)
((((B, E),D),C),A)
((((C, D),A),B), E)
((((C, D),A),E),B)
((((C, D),B),A),E)
((((C, D),B),E),A)
((((C, D),E),A),B)
((((C, D),E),B),A)
((((C, E),A),B),D)
((((C, E),A),D),B)
((((C, E),B), A),D)
((((C, E),B), D),A)
((((C, E),D),A),B)
((((C, E),D),B),A)
((((D, E),A),B),C)
((((D, E),A),C),B)
((((D, E),B),A),C)
((((D, E),B),C),A)
((((D, E),C),A),B)
((((D, E),C),B),A)
(((A,B), (C,D)), E)
(((A,C),(B,D)), E)
(((A,D),(B,C)), E)
(((A,B), (C,E)),D)
(((A,C),(B,E)),D)
(((A,E),(B,C)),D)
(((A,B), (D,E)),C)
(((A,D),(B,E)),C)
(((A,E),(B,D)),C)
(((A,C),(D,E)),B)
R71
R72
R73
R74
R75
R76
R77
R78
R79
R80
R81
R82
R83
R84
R85
R86
R87
R88
R89
R90
R91
R92
R93
R94
R95
R96
R97
R98
R99
R100
R101
R102
R103
R104
R105
(((A, D),(C,E)),B)
(((A, E),(C,D)),B)
(((B, C),(D,E)),A)
(((B, D),(C,E)),A)
(((B, E),(C,D)),A)
(((A, B),C),(D,E))
(((A, C),B),(D,E))
(((B, C),A),(D,E))
(((A, B),D),(C,E))
(((A, D),B), (C,E))
(((B, D),A),(C,E))
(((A, C),D),(B, E))
(((A, D),C),(B, E))
(((C, D),A),(B, E))
(((B, C),D),(A,E))
(((B, D),C),(A,E))
(((C, D),B),(A,E))
(((A, B),E),(C,D))
(((A, E),B),(C,D))
(((B, E),A),(C,D))
(((A, C),E),(B,D))
(((A, E),C),(B,D))
(((C, E),A),(B,D))
(((B, C),E),(A,D))
(((B, E),C),(A,D))
(((C, E),B), (A,D))
(((A, D),E),(B,C))
(((A, E),D),(B,C))
(((D, E),A),(B,C))
(((B, D),E),(A,C))
(((B, E),D),(A,C))
(((D, E),B),(A,C))
(((C, D),E),(A,B))
(((C, E),D),(A,B))
(((D, E),C),(A,B))
Page 24
24
Table 5 The 15 unrooted 5-taxon topological gene trees, as indicated by their non-trivial
splits, and their probabilities ui= P(Ti), where ri is the probability of the rooted gene tree
Rigiven the species tree σ+.
Tree
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
T13
T14
T15
SplitsProbability
AB|CDE, ABC|DE
AB|CDE, ABD|CE
AB|CDE, ABE|CD
AC|BDE, ABC|DE
AC|BDE, ACD|BE
AC|BDE, ACE|BD
AD|BCE, ABD|CE
AD|BCE, ACD|BE
AD|BCE, ADE|BC
AE|BCD, ABE|CD
AE|BCD, ACE|BD
AE|BCD, ADE|BC
BC|ADE, ABC|DE
BD|ACE, ABD|CE
BE|ACD, ABE|CD
u1= r1+ r2+ r59+ r60+ r67+ r76+ r105
u2= r3+ r4+ r53+ r54+ r64+ r79+ r104
u3= r5+ r6+ r47+ r48+ r61+ r88+ r103
u4= r7+ r8+ r57+ r58+ r70+ r77+ r102
u5= r9+ r10+ r41+ r42+ r65+ r82+ r101
u6= r11+ r12+ r35+ r36+ r62+ r91+ r100
u7= r13+ r14+ r51+ r52+ r71+ r80+ r96
u8= r15+ r16+ r39+ r40+ r68+ r83+ r95
u9= r17+ r18+ r29+ r30+ r63+ r94+ r97
u10= r19+ r20+ r45+ r46+ r72+ r87+ r89
u11= r21+ r22+ r33+ r34+ r69+ r86+ r92
u12= r23+ r24+ r27+ r28+ r66+ r85+ r98
u13= r25+ r26+ r55+ r56+ r73+ r78+ r99
u14= r31+ r32+ r49+ r50+ r74+ r81+ r93
u15= r37+ r38+ r43+ r44+ r75+ r84+ r90
Page 25
25
B 5-taxon unrooted gene tree distributions
B.1 Balanced species tree
For the 5-taxon balanced species tree of Fig. 2c,
σ+= (((a, b):x,c):y,(d,e):z),
let X = exp(−x), Y = exp(−y), and Z = exp(−z). Then the distribution of unrooted gene
trees Tiis given by ui= Pσ+(Ti) with
u1= 1 −2
3X −2
3Y Z −1
3X −1
3Y Z +1
3XY Z +
1
15XY3Z,
u2= u3=1
6XY Z −
1
10XY3Z,
1
15XY3Z,
1
10XY3Z,
u4= u13=1
3XY Z +
u5= u6= u9= u12=1
6XY Z −
u7= u8= u10= u11= u14= u15=
1
15XY3Z.
(11)
B.2 Rooted caterpillar species tree
For the 5-taxon rooted caterpillar species tree of Fig. 2d,
σ+= ((((a, b):x,c):y,d):z,e),
let X = exp(−x), Y = exp(−y), and Z = exp(−z). Then the distribution of unrooted gene
trees Tiunder the coalescent is given by ui= Pσ+(Ti) with
u1= 1 −2
3X −2
6XY −1
3Y −1
u4= u13=1
3Y +1
3XY +
1
18XY3+
1
90XY3Z6,
2
45XY3Z6,
1
90XY3Z6,
u2=1
3Y −1
9XY3+
1
18XY3−
u3=1
6XY −
3X −1
6XY −1
3XY +
1
18XY3+
1
90XY3Z6,
2
45XY3Z6,
1
90XY3Z6,
u5= u12=1
9XY3+
1
18XY3−
u6= u9=1
6XY −
u7= u8= u10= u11= u14= u15=
1
18XY3+
1
90XY3Z6.
(12)
B.3 Pseudocaterpillar species tree
For the 5-taxon pseudocaterpillar species tree of Fig. 2e,
σ+= (((a, b):x,(d,e):y):z,c),
let X = exp(−x), Y = exp(−y), and Z = exp(−z). Then the distribution of unrooted gene
trees Tiis given by ui= Pσ+(Ti) with
Page 26
26
u1= 1 −2
3X −2
3Y +4
5
18XY +
5
18XY +
9XY −
2
45XY Z6,
u2= u3=1
3Y −
1
90XY Z6,
1
90XY Z6,
u4= u13=1
3X −
u5= u6= u7= u9= u10= u12= u14= u15=
1
18XY +
1
90XY Z6,
u8= u11=1
9XY −
2
45XY Z6.
(13)
C Nonbinary species trees
Proof (of Proposition 11) The extension of Proposition 3 to nonbinary σ+was discussed in
Section 5.
From this, for |X| ≥ 5 we know that for Q ⊂ X with |Q| = 4, the possibly unresolved
unrooted quartet tree on Q can be determined from gene tree probabilities. Thus the unrooted,
labeled species tree σ−can be determined by the identifiability of (possibly nonbinary) phylo-
genetic trees from their quartets (Bandelt and Dress 1986)(Semple and Steel 2003, Theorem
6.3.5), and thus Corollary 6 has been extended.
Next, in addition to the three fully resolved rooted tree shapes on 5 taxa, we must consider
the nine rooted shapes with polytomies. In Table 6, we designate these as P1,...,P9, specify
an arbitrary labeling of the leaves of each, and list inequalities analogous to inequalities (4)–
(6) for unrooted gene tree probabilities. The equivalence classes of labeled, binary, unrooted
5-taxon gene trees associated with each polytomous species tree are given in Table 7, along
with the gene tree probabilities as functions of transformed branch lengths X, Y , and Z. Gene
tree probabilities are obtained from the equations for resolved trees in Appendix B by setting
one or more branch lengths to 0.
For the 5-taxon species tree shapes, in all cases of either resolved and polytomous trees,
the least probable class, C, of gene trees always has probability strictly smaller than all others.
There are five possible cases for the cardinality of C:
1. |C| = 15: polytomy P1
2. |C| = 12: polytomy P2 or polytomy P3
3. |C| = 10: polytomy P5 or polytomy P7
4. |C| = 8: resolved pseudocaterpillar
5. |C| = 6: resolved caterpillar, resolved balanced, polytomy P4, polytomy P6, polytomy P8,
or polytomy P9
If |C| = 12, we can distinguish between P2 and P3 since all gene trees in the 3-element
class for P3 have the same taxon not occurring in a cherry, while for P2 the gene trees in the
3-element class have different taxa in this role.
For |C| = 10, we can distinguish between polytomies P5 and P7 by considering the two
2-element classes for both. For P5, both of the classes {T2,T3} and {T4,T13} contain trees
with one cherry in common. For P7, the gene trees in the classes {T2,T3} have a cherry in
common, but those in {T8,T11} do not. Note that it is possible for these classes to degenerate,
to form a 4-element class, but by counting the number of trees with a cherry in common in
the larger degenerate class we can still determine whether the species tree shape is P5 or P7.
If |C| = 6 the cardinality of the class with the second smallest probability determines the
rooted tree shape in some cases. The class with the second smallest probability has cardinality
2 only for the resolved caterpillar, cardinality 3 only for P9, cardinality 6 only for polytomies
P4 and P8, and cardinality 4 for the resolved balanced tree and P6.
At this point we have determined the rooted unlabeled topology of the species tree from
the 5-taxon gene tree classes, except for the P4 versus P8 case and the balanced versus P6
case. (We will return to these cases later.)
For the fully resolved trees, Proposition 8 explains how we determine the labeling, so similar
arguments are needed for each polytomous tree. If the species tree is P1, there is nothing to
Page 27
27
Table 6 Representatives for the 9 nonbinary, rooted 5-taxon species tree shapes, with in-
equalities for gene tree probabilities.
Species treeNewick representative inequalities species tree shape
P1
(a, b,c,d,e)
P2
(a, b,c,(d,e):z)
u1> u2
P3
((a, b,c,d):z,e)
u3> u1
P4
((a, b,c):y,d,e)
u1> u2> u7
P5
((a, b):x,(d,e):y,c)
u1> u2,u4> u5
P6
(((a, b):x,c):y,d,e)
u1> u2,u4> u5> u7
P7
((a, b):x,d,e):z,c)
u1> u2,u8> u4
P8
(((a, b,c):y,(d,e):z)
u1> u2> u7
P9
(((a, b,c):y,d):z,e)
u1,u3> u2> u7
do. For polytomies P2, P5, and P4/P8 the labeling on the unrooted tree determines that on
the rooted one.
If the species tree is P3, the taxon that appears in no cherry in the gene trees in the
3-element class is the one that is an outgroup to all the others in the species tree.
For polytomy P7, the resolved cherry in the species tree is determined by the unrooted
labeled tree, and the outgroup is determined by not appearing in a cherry in the most probable
gene tree.
For polytomy P9, the non-outgroup taxon which is not descended from the polytomy in
the species tree is distinguished by not appearing in any cherry in the class with the second
smallest probability. Calling this identified taxon d, the outgroup taxon is determined as the
one appearing in a cherry with d in three of the six most probable trees (i.e., in three trees in
the union of the two most probable classes, which may degenerate to a single class).
Finally, the labeling on the balanced/P6 tree is determined as it was for the balanced tree
in the proof of Proposition 7.
At this point we have determined the labeled rooted species tree topology ψ+(except for
two P4/P8 and balanced/P6 ambiguities).
It remains to determine branch lengths on ψ+. If the unlabeled species tree is any tree
other than the resolved balanced tree, P4, P6, or P8, then the branch lengths can be solved
from the system of equations listed for the given species tree from Appendix B or the formulas
in Table 7.
If the species tree is known to be either P4 or P8, we note that P8 degenerates to P4 as
z → 0 (or Z → 1). Solving the system of equations for the uis in terms of the branch lengths
for P8, if Z < 1, then the species tree is P8. If Z = 1, then the species tree is P4. Similarly,
P6 is the limiting case of the balanced 5-taxon species tree as Z → 1. Solving the system of
equations for the balanced tree, the species tree is the balanced tree if Z < 1 and is P6 if
Z = 1.
Thus, for a 5-taxon species tree, even with polytomies, σ+is identifiable. This extends
Propositions 7 and 8 to potentially nonbinary species trees. Theorem 9 also extends, noting
that if the root of the species tree has degree greater than 2, then its location will be identified
by some 5-taxon subtree with the same property. The proof of Corollary 10 did not use the
assumption that σ+is binary, so it applies to nonbinary species trees as well. ⊓ ⊔
Page 28
28
Table 7 Equivalence classes of equiprobable gene trees for each nonbinary 5-taxon species
tree, and gene tree probabilities in terms of transformed branch lengths. T denotes the set of
all 5-taxon binary unrooted gene trees, {T1,T2,...,T15}.
Species treeequivalence classes gene tree probabilities
P1
T
ui=
1
15
P2
{T1,T4,T13}
u1=1
3−
1
15Z
4
15Z
T \ {T1,T4,T13}
u2=
P3
{T3,T6,T9}
u3=1
9−
1
18+
3−1
2
45Z6
1
90Z6
T \ {T3,T6,T9}
u1=
P4
{T1,T4,T13}
u1=1
3Y +
1
10Y3
1
15Y3
{T2,T3,T5,T6,T9,T12}
u2=1
6Y −
1
15Y3
{T7,T8,T10,T11,T14,T15}
u7=
P5
{T1}
u1= 1 −2
3X −2
4
15XY
4
15XY
3Y +2
5XY
{T2,T3}
u2=1
3Y −
{T4,T13}
u4=1
3X −
1
15XY
T \ {T1,T2,T3,T4,T13}
u5=
P6
{T1}
u1= 1 −2
3X −2
3Y +1
3XY +
10XY3
1
15XY3
1
15XY3
{T2,T3}
u2=1
3Y −1
3X −1
6XY −
1
{T4,T13}
u4=1
3XY +
1
10XY3
{T5,T6,T9,T12}
u5=1
6XY −
1
15XY3
3−2
{T7,T8,T10,T11,T14,T15}
u7=
P7
{T1}
u1=1
9X −
5
18X +
2
45XZ6
1
90XZ6
3−1
u2=1
6Y Z −
u7=
15Y3Z
u1=1
2
45XZ6
1
90XZ6
{T2,T3}
u2=1
3−
{T8,T11}
u8=1
9X −
1
18X +
T \ {T1,T2,T3,T8,T11}
u4=
P8
{T1,T4,T13}
u1=1
3Y Z +
10Y3Z
1
15Y3Z
{T2,T3,T5,T6,T9,T12}
1
{T7,T8,T10,T11,T14,T15}
1
P9
{T1,T4,T13}
3−1
6Y −1
3Y +
9Y3+
1
18Y3−
18Y3+
1
18Y3+
1
90Y3Z6
2
45Y3Z6
90Y3Z6
1
90Y3Z6
{T2,T5,T12}
u2=1
{T3,T6,T9}
u3=1
6Y −
1
{T7,T8,T10,T11,T14,T15}
u7=
1
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