Syst. Biol. 47(4):617–624, 1998
Using a Nonrecursive Formula to Determine Cladogram Probabilities
J. STONE1, 3 AND J. REPKA2
Department of Zoology, University of Toronto, Toronto, Ontario M5S 3G5, Canada;
E-mail: [email protected]
2
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada;
E-mail: [email protected]
1
Abstract.— Three properties of bifurcating branching diagrams that are used for representing a
speciŽc number of taxa are (1) the number of possible arrangements, (2) the number of possible
topologies, and (3) the probabilities of formation according to particular models of cladogenesis.
Of these, the probabilities have received the least attention in the literature. Indeed, many biologists would be astonished by the observation that the probability of a commonly cited cladogram
containing 35 phyla of the animal kingdom is < 0.0072% of the value of the average probability
taken over all possible cladograms! We reviewed works on cladogram arrangements and topologies
and developed a computer-generated table of enumerations that extends and corrects such tables
in the literature. We also developed a nonrecursive formula for the determination of cladogram
probabilities. This formula facilitates calculation and thereby should promote use of cladogram
probabilities, which might provide more accurate null hypotheses for tests of cladogenic events
than do considerations of cladogram arrangements or topologies. [Bifurcating branching diagram;
cladogram arrangement; cladogram topology; likelihood; null hypothesis; phylogenetic tree.]
An important question confronts every
systematist who performs a cladistic analysis: How many distinct cladograms (bifurcating branching diagrams) representing n
taxa are possible? Many researchers (Cayley, 1856, 1889; Schr öder, 1870; Edwards and
Cavalli-Sforza, 1964; Cavalli-Sforza and Edwards, 1967; Moon, 1970; Harding, 1971;
Dobson, 1974; Harper, 1976; Phipps, 1976a,
1976b; Felsenstein, 1978) have considered
various mathematical versions of this question, have provided methods for enumerating the number of distinct bifurcating or
multifurcating branching diagrams consisting of n labeled terminal nodes and labeled
or unlabeled nonterminal nodes, and have
tabulated values. An important distinction
considered by only a few researchers (Harding, 1971; Phipps, 1976a; Simberloff et al.,
1981; Savage, 1983; Slowinski and Guyer,
1989) is the distinction between cladogram arrangement and cladogram topology. For example, three taxa can be arranged into three distinct cladogram arrangements, but there is
only one cladogram topology. Enumerations
of cladogram arrangements and topologies
are presented in Table 1, which extends and
corrects such tables in the literature. A third
cladogram property, which has received less
3
Present address (and address for correspondence):
Department of Ecology and Evolution, State University
of New York, Stony Brook, NY 11794-5245, USA.
attention in published works than has arrangement or topology, concerns probability. Cladogram probability refers to the likelihood of obtaining a particular bifurcating
branching diagram according to a particular
model of cladogenesis.
In this paper, formulae for the enumeration of cladogram arrangements and
topologies are reviewed and a nonrecursive
formula for calculating cladogram probabilities is presented. The simple, nonrecursive
formula is more amenable to computer programming and so will facilitate and increase
the efŽciency of calculating cladogram probabilities.
CLADOGRAM ARRANGEMENT
The number of distinct rooted bifurcating
branching diagrams with n labeled terminal
nodes can be calculated by using the formula
P (2i – 3),
(1)
where the product runs between i = 2 and n
(Edwards and Cavalli-Sforza, 1964; CavalliSforza and Edwards, 1967; Moon, 1970;
Harding, 1971; Phipps, 1976a; Felsenstein,
1978). To simplify calculations, the formula
usually is presented (Harding, 1971; Phipps,
1976a; Felsenstein, 1978) in the form
617
(2n – 3)!/ [2n–2 (n – 2)!].
(2)
618
TABLE 1.
Numbers of cladogram arrangements and topologies representing n taxa.
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
VOL. 47
SYSTEMATIC BIOLOGY
Arrangements
Topologies
1
1
3
15
105
945
10,395
135,135
2,027,025
34,459,425
654,729,075
13,749,310,575
316,234,143,225
7,905,853,580,625
213,458,046,676,875
6,190,283,353,629,375
191,898,783,962,510,625
6,332,659,870,762,850,625
221,643,095,476,699,771,875
8,200,794,532,637,891,559,375
319,830,986,772,877,770,815,625
13,113,070,457,687,988,603,440,625
563,862,029,680,583,509,947,946,875
25,373,791,335,626,257,947,657,609,375
1,192,568,192,774,434,123,539,907,640,625
1
1
1
2
3
6
11
23
46
98
207
451
983
2,179
4,850
10,905
24,631
56,011
127,912
293,547
676,157
1,563,372
3,626,149
8,436,379
19,680,277
In more recent publications concerning bifurcating branching diagram enumeration
(Swofford et al., 1996:410; Trueman, 1993),
various authors have provided the formula
for enumerating the number of distinct unrooted bifurcating branching diagrams with
n labeled terminal nodes:
P (2i – 5),
(3)
where the product runs between i = 3
and n.
W(n) =
T(i) T(n–i) / 2 + E(n) ,
(5)
where the sum runs between i = 1 and (n–1),
E(n) = 0, if n is odd, or W(n/ 2) / 2 if n is even,
and W(1) = 1.
CLADOGRAM PROBABILITY
CLADOGRAM TOPOLOGY
The number of distinct bifurcating
branching diagram topologies with n terminal nodes can be enumerated by using a
two-term recursive formula (Phipps, 1976a):
T(i) T(n–i) if n is odd, or
T(i) T(n–i) + [T(n/ 2) (T(n/ 2) + 1)/ 2],
if n is even,
j terminal nodes and the summation is performed over all integers i satisfying 1 #
i < n/ 2. A simpler but equivalent recursive
enumeration was provided by Wedderburn
(1922):
(4)
in which T( j) is the number of topologically
distinct bifurcating branching diagrams for
A natural extension of the preceding
works is the calculation of probabilities of
constructing the various distinct bifurcating
branching diagrams under the assumption
that there is an equal likelihood of bifurcating (or terminating) at every node during the
(Markovian) construction of the diagrams.
Harding (1971: Formula 4.1) provided a recursive formula for the calculation of such
probabilities. SpeciŽcally, for a bifurcating
branching diagram with n terminal nodes, if
the two clades that originate from the most
basal node yield n0 and n0 0 terminal nodes
1998
STONE AND REPKA—CLADOGRAM PROBABILITIES
and the probabilities of constructing these
clades are p0 and p0 0 , respectively, then the
probability of obtaining that particular diagram among all distinct bifurcating branching diagrams with n(= n0 +n0 0 ) terminal nodes
is
2(n – 1)–1 p0 p0 0 ,
if the topologies of the two clades are distinct, or
(n – 1)–1 p0 p0 0 = (n – 1)–1 (p0 )2 ,
(6)
if the topologies of the two clades are identical. This formula can be used recursively
to calculate the probability of constructing
any cladogram, beginning at the most basal
node and continuing toward the terminal
nodes.
It is natural to distinguish cases in which
the topologies of two clades originating
from a nonterminal node are identical or distinct. For a nonterminal node u, the function d (u) (modeled after the Kroenecker delta
function) is deŽned to be 0, if the topologies
of the two clades originating from u are distinct, or 1, if they are identical. Then formula
6 can be written as
[2 – d (u0 )](n – 1)–1 p0 p0 0 ,
(60 )
in which u0 represents the most basal node.
Formula 6 can be used to derive a nonrecursive formula that is easy to use. The probability of constructing a cladogram with a
particular topology (among all those with n
terminal nodes) is
(1/ (n – 1)!)P C(n1 (u) + n2 (u) – 2,
n1 (u) – 1)(2 – d (u)),
(7)
where the product is over all nonterminal
nodes u, C denotes the combinatorial function (i.e., C(n, k) = n!/ [k!(n – k)!]), and n1 (u)
and n2 (u) are the numbers of terminal nodes
contained in the two clades originating from
u (a proof is provided in the Appendix).
REMARKS
The numbers exhibited adjacent to the
cladograms in Figure 1 are the numerators
of the probabilities obtained with use of
619
formula 7 (i.e., omitting the factor 1/ (n –
1)!). Dots in the Žgure indicate nonterminal
nodes from which two identical clades originate (i.e., nonterminal nodes with d (u) = 1).
With a little practice, probabilities are easy
to calculate.
If cladogram topologies are arranged in
order of increasing probability (n = 2 to 9,
Fig. 1), some interesting generalizations can
be made: For any particular value of n, (1)
the least probable cladogram topology is the
one that contains terminal sister group pairs,
themselves sister to a completely pectinate
branching sequence of (n – 4) terminal
nodes; (2) (a corollary of 1) the completely pectinate cladogram never is the least
probable—in fact, it is twice as probable as
the least-probable topology; and (3) cladograms can be partitioned according to the
numbers of terminal nodes contained in
each of the two clades originating from the
most-basal node (k and n – k, for any integer k satisfying 1 # k # n/ 2). As has been
shown (Feller, 1968; Slowinski and Guyer,
1989; Heard, 1992), the total probability of
all cladograms in any such partition equals
2/ (n – 1) except that, when n is even, the
cladograms with clades containing (n/ 2)
and (n/ 2) terminal nodes have total probability 1/ (n – 1).
A cladogram having a low probability
represents a bifurcating branching diagram
that can be constructed in very few ways.
The cladograms having the lowest probabilities generally are highly asymmetric; however, some highly symmetric cladograms
also have comparatively low probabilities.
For example, whenever n = 2k , for any k,
there is a very symmetric diagram in which
every nonterminal node joins two identical
clades (such nonterminal nodes are indicated by dots in Fig. 1) and every terminal
node is exactly k nonterminal nodes away
from the root node (Fig. 1: k = 1, 2, 3; i.e.,
n = 2, 4, 8). Such diagrams have low probabilities. Both highly asymmetric and highly
symmetric diagrams display a rigidity that
makes it difŽcult to introduce much variety into the process by which they are
constructed.
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SYSTEMATIC BIOLOGY
VOL. 47
FIGURE 1. Cladograms with n terminal nodes for n = 2, . . . , 9. For each value of n, cladograms are presented
in order of increasing probability. Numbers adjacent to the cladograms are the numerators of the corresponding
probabilities if they are expressed as fractions with denominator (n – 1)! (for each n, the numerators, counted
with repetitions, sum to (n – 1)!). Dots in the Žgure indicate nonterminal nodes from which two identical clades
originate.
AN EXAMPLE
An intriguing exercise that yields an astonishing result is the application of formula
7 to a particular cladogram that has been inferred to represent the phylogeny of the animal kingdom (Meglitsch and Schram, 1991:
Fig. 38.2; and Fig. 2, this paper). The probability of this particular hypothesis of meta-
zoan evolution equals » (1/ 13912) times
the value of the average probability taken
over all possible cladogram topologies representing 35 phyla! The implications of this
small value are either that (1) this particular
hypothesized scenario of animal evolution
was a very improbable event in the history of
life on this planet (if cladogenesis truly has
been Markovian and the cladogram accu-
1998
STONE AND REPKA—CLADOGRAM PROBABILITIES
621
FIGURE 2. Cladogram of the Metazoa (adapted from Meglitsch and Schram, 1991: Fig. 38.2); Placozoa and
Mesozoa are excluded. The nonterminal node immediately preceding that from which Sipunculida and Echiura
originate originally consisted of a trichotomy comprising the clades Sipunculida + Echiura, Mollusca, and the
taxa originating from the nonterminal node at the base of the Annelida. Numbers adjacent to labeled nonterminal
nodes are factors corresponding to the product of formula 7. Dots in the Žgure indicate nonterminal nodes from
which two identical clades originate. Each of these contributes a factor of 1 to the product of formula 7, except
for the one that is labeled, which contributes a factor of 12. Each of the other unlabeled nonterminal nodes
in the Žgure contributes a factor of 2. The probability of this cladogram is 2(2)2(2)2(6)12(2)11,705,850(42)2(2)2
(2)2(2)2(151,164)2(2)2(18)2(2)2(2)2/34! = 1/1,461,601,812,825,000. Using formula 4 or 5, there are 105,061,603,969
cladogram topologies representing 35 taxa, so the average probability is 1/105,061,603,969,which is approximately
13,912/1,461,601,812,825,000. Thus, the probability of this cladogram of the Metazoa is 1/13 912 or 0.00719% of
the value of the average probability taken over all possible cladograms. Less conservative, smaller estimates
of » 0.0064% or 0.00080% result if the clades Mollusca and Sipunculida + Echiura originate from a common
nonterminal node or if the clades Sipunculida + Echiura and the taxa originating from the nonterminal node at
the base of the Annelida originate from a common nonterminal node, respectively.
622
SYSTEMATIC BIOLOGY
rately reconstructs the history of metazoan
evolution) or (2) the assumption of Markovian cladogenesis is unrealistic. There is no
reason to favor implication (1); however, evidence has been presented to support arguments that rejection of this analysis because
of implication (2) might be either too drastic (Savage, 1983) or justiŽed (Guyer and
Slowinski, 1991, 1993; Heard, 1992; Mooers, 1995). In either case, this cladogram is
presented for illustrative purposes only—
its terminal nodes represent phyla, which
cannot speciate (i.e., it is incomplete [Mooers, 1995; Mooers and Heard, 1997])—to
demonstrate how a cladogram probability
can be used to establish a null hypothesis. Nevertheless, cladogram probabilities
should be considered cautiously, because
implicit in the Markovian model are many
assumptions, some of which might be inappropriate in particular instances (Losos
and Adler, 1995; Mooers and Heard, 1997).
For example, different taxa live in different
environments, contain different numbers of
species or individuals, and undergo different rates of mutation and rates and modes
of speciation; in a Markovian model, however, all such factors are assumed to be of
no consequence.
DISCUSSION
Enumerating the cladogram arrangements possible for a number of taxa is
requisite to determining the likelihood of
obtaining stochastically a bifurcating
branching diagram with particular sister
group relationships (equal to the reciprocal of the values obtained by using formula 1 or formula 3) and thereby provides
a measure of conŽdence of that particular hypothesis of evolutionary relationships
(similar to the use of the proportional-todistinguishable-arrangements null hypothesis of Simberloff et al. [1981], HD ). Enumerating the cladogram topologies possible
for a speciŽed number of taxa is requisite
to determining the likelihood of obtaining a particular bifurcating branching diagram topology under the assumption that
all topologies representing that number of
taxa are equiprobable and thereby provides
a measure of conŽdence of that particular
classiŽcation scheme (similar to the use of
VOL. 47
the equiprobable null hypothesis of Simberloff et al. [1981], HE ). Calculating a particular cladogram probability is requisite to
determining the likelihood of obtaining a
particular hierarchy of classiŽcation under
the assumption of a Markovian bifurcating
branching process and thereby provides a
measure of conŽdence of those particular
cladogenic events (similar to the use of the
Markovian branching null hypothesis of
Simberloff et al. [1981], HM ). Therefore, the
enumeration of cladogram arrangements
and topologies concerns the diagrams (or
patterns) resulting from (the process of)
cladogenesis, whereas the calculation of
cladogram probability more directly concerns cladogenesis itself (Savage, 1983); the
nonrecursive formula also can be applied
to gene trees within populations (Maddison
and Slatkin [1991], for example).
Felsenstein (1978) concluded the abstract
of his paper by stating that one of the principal uses of enumerating cladogram arrangements was to frighten taxonomists. In the
same spirit, the determination of cladogram
probabilities might assist frightened systematists with their interpretations of their
works: The quantiŽcation of certain vague
concepts, such as the diversity of clades
contained in cladograms, might be facilitated by measures of probability. For example, given a particular hierarchical classiŽcation of n taxa containing a taxon-rich
clade, the probabilities of other distinct hierarchical classiŽcations of n taxa that include
a clade with at least as many taxa can be
determined. Hitherto, such quantiŽcations
have consisted of visual comparisons of diversity diagrams of real clades with those
of clades generated by stochastic branching models (Raup et al., 1973; Gould et
al., 1977), calculations of numerical indices
of balance (Shao and Sokal, 1990; Heard,
1992), examinations of distributions of taxonomic subunits within taxonomic units
(Dial and Marzluff, 1989), interpretations of
fractal dimensions as indicators of diversity (Burlando, 1990), utilization of stratigraphic or molecular dating methods (Hey,
1992; Nee et al., 1992), comparisons of cladogram topologies with nonparametric distributions of computer-simulated topologies
(Kirkpatrick and Slatkin, 1993), and consid-
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STONE AND REPKA—CLADOGRAM PROBABILITIES
erations of bifurcating branching diagrams
(Farris, 1976; Slowinski and Guyer, 1989).
It is hoped that the nonrecursive formula
presented in this paper will increase the
utility of cladogram probabilities, provoke
discussion, promote additional discoveries,
and thereby ultimately eventually provide a
basis for selecting among equally parsimonious but distinct competing hypotheses.
ACKNOWLEDGMENTS
Concerning this work, M. Telford provided calculated commentary; S. B. Heard, W. P. Maddison, and J. B.
Slowinski provided informative, detailed, and thoughtprovoking reviews; and D. Cannatella recursively provided editorial suggestions and encouragement.
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Received 10 April 1997; accepted 2 February 1998
Associate Editor: D. Cannatella
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APPENDIX. PROOF OF FORMULA 7
The proof is by mathematical induction. It is supposed that the result is known for all patterns with
fewer than n nonterminal nodes. In particular, the result
is known for the two clades originating from the most
basal node u0 , which have n0 and n0 0 terminal nodes
(n = n0 + n0 0 ). This means that the probabilities of constructing these clades are
p0 = (1/ (n0 – 1)!)P C(n1 (u0 )
+ n2 (u0 ) – 2, n1 (u0 ) – 1)(2 – d (u0 )),
p0 0 = (1/ (n0 0 – 1)!)P C(n1 (u0 0 )
+ n2 (u0 0 ) – 2, n1 (u0 0 ) – 1)(2 – d (u0 0 )),
p = (2 – d (u0 ))(n – 1)–1 (1/ (n0 – 1)!)(1/ (n0 0 – 1)!)
´ P C(n1 (u) + n2 (u) – 2, n1 (u) – 1)(2 – d (u))
= (2 – d (u0 ))((n – 1)!)–1 ((n – 1)!/ (n – 1))
(1/ (n0 – 1)!)(1/ (n0 0 – 1)!)
´ P C(n1 (u) + n2 (u) – 2, n1 (u) – 1)(2 – d (u))
(80 )
(80 0 )
where products are performed over the nonterminal
nodes (u0 and u0 0 ) in each branch. Formula 60 can be
used to show that the probability of constructing the
entire cladogram is
p = (2 – d (u0 ))(n – 1)–1 p0 p0 0
= (2 – d (u0 ))(n – 1)–1
´ (1/ (n0 – 1)!)P C(n1 (u0 ) + n2 (u0 )
– 2, n1 (u0 ) – 1)(2 – d (u0 ))
´ (1/ (n0 0 – 1)!)P C(n1 (u0 0 ) + n2 (u0 0 )
– 2, n1 (u0 0 ) – 1)(2 – d (u0 0 )).
form a product over every nonterminal node in the
whole cladogram except the most basal node u0 . So
= (2 – d (u0 ))((n – 1)!)–1
((n – 2)!/ ((n0 – 1)!(n0 0 – 1)!))
´ P C(n1 (u) + n2 (u) – 2, n1 (u) – 1)(2 – d (u))
= (2 – d (u0 ))((n – 1)!)–1 C(n – 2, n0 – 1)
´ P C(n1 (u) + n2 (u) – 2, n1 (u) – 1)(2 – d (u)),
(9)
where the product runs over all nonterminal nodes except u0 . Recalling that n = n0 + n0 0 ,
p = (1/ (n – 1)!)P C(n1 (u)
+ n2 (u) – 2, n1 (u) – 1)(2 – d (u)),
0
(6 )
In this formula, the two products are performed over
the nonterminal nodes in the two clades. Together they
where the product now runs over all nonterminal
nodes. This is formula 7.
The initial step in the induction is trivial, because
there is only one topology for a bifurcating branching
diagram with two terminal nodes, that is, its probability
is 1, so the proof is complete.