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Subject: [Learn - LIU Comp Sci] This is an interesting treatise on Graph
Theory from Palaeontology
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This is an interesting treatise on Graph Theory from Palaeontology
Are node-based and stem-based clades equivalent? Insights from graph theory
November 18, 2010 =C2=B7 Tree of Life =
Print or Save PDF
E. O. Wiley
Despite the prominence of =E2=80=9Ctree-thinking=E2=80=9D among contemporary
systematists and evolutionary biologists, the biological meaning of
different mathematical representations of phylogenies may still be
muddled. We compare two basic kinds of discrete mathematical models used
to portray phylogenetic relationships among species and higher taxa:
stem-based trees and node-based trees. Each model is a tree in the sense
that is commonly used in mathematics; the difference between them lies
in the biological interpretation of their vertices and edges. Stem-based
and node-based trees carry exactly the same information and the
biological interpretation of each is similar. Translation between these
two kinds of trees can be accomplished by a simple algorithm, which we
provide. With the mathematical representation of stem-based and
node-based trees clarified, we argue for a distinction between types of
trees and types of names. Node-based and stem-based trees contain
exactly the same information for naming clades. However, evolutionary
concepts, such as monophyly, are represented as different mathematical
substructures in the two models. For a given stem-based tree, one should
employ stem-based names, whereas for a given node-based tree, one should
use node-based names, but applying a node-based name to a stem-based
tree is not logical because node-based names cannot exist on a
stem-based tree and visa versa. Authors might use node-based and
stem-based concepts of monophyly for the same representation of a
phylogeny, yet, if so, they must recognize that such a representation
differs from the graphical models used for computing in phylogenetic
JLM was partially supported by NSA Young Investigators Grant
#H9B230-08-1-0073. DCB Partial support to EOW by NSF DEB 073289, the
Euteleost Tree of Life project, which includes a component aimed at
increasing understanding of phylogenetic trees, is gratefully acknowledged.
=E2=80=9CTree-thinking=E2=80=9D, using phylogenies to understand evolutiona=
relationships, name clades, and understand evolutionary transformations
and biogeography, is now ubiquitous in systematics and evolutionary
biology and is making its way quickly into the educational and public
realms (e.g.,  ;  ; ). But the biological interpretation of the
precise mathematical notion of a tree often remains unclear (). We
argue that the two dominant representations of phylogenies used today
(node-based and stem-based) are mathematically equivalent, but not
identical. We then argue that if these two forms of trees are not
considered separate and distinct representations of the same
information, then biological interpretations of trees and evolutionary
transformations may become confused.
In the Willi Hennig Memorial Symposium, held in 1977 and published in
Systematic Zoology in 1979, David Hull expressed the concern that
=E2=80=9Cuncertainty over what it is that cladograms are supposed to depict=
how they are supposed to depict it has been one of the chief sources of
confusion in the controversy over cladism=E2=80=9D (, p.420). Early
disagreements concerning the differences between cladograms and
phylogenetic trees were largely generated by such differences (; ;
;   ). This debate has largely subsided, yet the
importance of representing phylogenies and interpreting their biological
meaning remains. The purpose of this article is to compare what we
believe to be the two most commonly used tree models of phylogenetic
relationships, namely node-based and stem-based (or branch-based) trees,
using the mathematical techniques of graph theory. We consider
node-based and stem-based trees to be representations of phylogenies as
both explicitly model hypotheses of common ancestry. We assert that it
is imperative to understand the mathematical relationships between these
two graphical representations of phylogenies to make meaningful
biological statements. In doing so, we aim to finally lay to rest the
=E2=80=9Cuncertainty=E2=80=9D observed by Hull thirty years ago.
The vertices of a node-based tree represent taxa (sampled or inferred),
while its edges model ancestry relationships. For example, if the tree
represents the results of a phylogenetic analysis, then the tips of the
tree are nodes and internal nodes represent inferred common ancestors.
By contrast, in a stem-based tree, both sampled and inferred ancestral
taxa are modeled by edges, while vertices correspond to speciation
events. These two models are isomorphic (as that term is used in
mathematics) but not equal: that is, they carry exactly the same
information about ancestry, but it is encoded in two different ways. To
make this explicit, we give a simple algorithm that constructs a unique
node-based tree for every stem-based tree and vice versa. While some
might see as frivolous the demonstration that these two tree models are
equivalent, the relationship between these two representations has
important repercussions for evaluating the biological meaning of trees.
Thus, we provide an explicit example of the need for distinction between
these representations through a discussion of how the phylogenetic
concept of monophyly is represented in each graphical model.
Some basic graph theory
Mathematically speaking, all of the diagrams we shall consider are
graphs: they are finite structures built out of vertices (sometimes
called nodes) and edges, in which each edge connects two vertices (see
) for background. A graph is usually represented by drawing the
vertices as dots and the edges as line segments. Frequently, the
vertices and/or edges are labeled with names, numbers, or other data.
Graphs provide a simple and powerful tool to model and study
phylogenetic and synapomorphic relationships between taxa (and many
other structures). Utilizing graphs as representations of this sort has
a long history in the study of organismal evolution with famous early
examples including the sole figure in Charles Darwin=E2=80=99s () Origi=
Species. However, one must be very careful to keep track of what the
individual vertices and edges are supposed to mean, particularly when
there is more than one way to represent the same biological data in a
graph. Until the techniques promoted by Hennig () gained wide use,
graphs were essentially cartoons sketched out by hand rather than
representing the output of an analytical inference in the sense that
phylogenies are now typically used. With the advent of phylogenetic
analyses, the representation used for trees of evolutionary
relationships became non-trivial. Before proceeding, we mention a few
basic facts and terms from graph theory, so as to have a unified
mathematical language with which to work. We will introduce more
technical material later, as needed.
We will primarily be concerned with graphs that are trees.
Mathematically, a tree is a graph T containing no closed loops;
intuitively, if you walk along the edges from vertex to vertex, the only
way to return to your starting point is to retrace your steps. Put in an
evolutionary context, this means that trees in this sense cannot have
reticulations within them. If we designate one vertex r as the root of
T, then every edge connects a vertex x that is closer to r with a vertex
y that is further away. In this case, we say that x is the parent of y,
and it is often convenient to regard the edge between them as a directed
edge (or arc ) pointing from x to y, represented by the symbol x =E2=86=92 =
Every vertex in a tree has a unique parent, except for the root, which
has no parent. An immediate consequence is the useful fact that every
tree with n edges has n +1 vertices, and vice versa, though, of course,
several different vertices may share a common parent (i.e., a polytomy).
The ancestors of a vertex are its parent, its parent=E2=80=99s parent, its
parent=E2=80=99s parent=E2=80=99s parent, and so on. Equivalently, we might=
say that an
edge x=E2=86=92y is an ancestor of another edge a =E2=86=92 b if y is equal=
to, or an
ancestor of, a . A lineage (or ancestral lineage) of a vertex x is the
complete list of vertices that are ancestors of x and are descendants
of, or equal to, some other vertex y. If y =3D root(T), then this list is
called the total lineage of x. It is important to note that in a tree
with a root the choice of a root vertex, together with the topology of
the tree, completely determines all ancestry relationships.
A subtree of a tree T is a tree U all of whose vertices and edges are
vertices and edges of T as well. This is equivalent to saying that U can
be formed by removing some vertices and edges from T. If in addition T
is a rooted tree, then U inherits its =E2=80=9Cancestor-of=E2=80=9D relatio=
n from T as
well. A proper subtree of a rooted tree is a subtree that consists of a
vertex and all its descendants. A proper subtree is uniquely determined
by its root vertex, so there are exactly as many proper subtrees of T as
there are vertices.
Trees are well suited for modeling phylogenetic relationships between
species or taxa, in which each species or taxon has a unique parent.
Uniqueness is vital; a tree in the sense that we use it here cannot
model reticulations, such as tokogenetic relationships in a sexually
reproducing species or hybridization events between two different species.
Fig. 1: An example of a stem-based tree, indicating the evolutionary
relationship among the sampled taxa A, B, C and their unsampled, but
inferred, ancestral species y and z.
A. An example of a stem-based tree, indicating the evolutionary
relationship among the sampled taxa A, B, C and their unsampled, but
inferred, ancestral species y and z. =E2=80=94 (B) The same tree with chara=
data shown (the names of the internal edges have been omitted for
clarity). In each case, taxon names are displaced from the leaf position
to emphasize that the edge is the taxon.
By the term stem-based tree, we mean a tree that models (hypothesized)
phylogenetic relationships among taxa by depicting taxa as edges and
speciation events as vertices. For instance, in the tree in Fig. 1A, the
terminal edges, labeled A, B, and C, represent named taxa; that is,
larger groups of individual organisms represented by sampled specimens.
The internal edges, labeled y and z, represent ancestral lineages needed
to account for the terminal taxa under the paradigm of descent with
modification. The vertices represent speciation events, in which the
edge below the vertex is the common ancestor and the edges above it are
descendants. Mathematically, the edge y is the youngest common ancestor
of edges B and C . Biologically, moving up the tree represents moving
forward in time, so the edge y represents a lineage of common ancestors
of the sampled taxa Band C, occurring before the speciation event that
distinguishes B and C and after any previous speciation events. Thus the
total lineage of a species (or, more properly, a hypothesis of its
lineage) is represented by a chain of edges starting with the species
itself and moving down the tree towards the root vertex, which
necessarily has only one edge emanating from it=E2=80=94representing the co=
ancestor of all sampled taxa.
We frequently refer to the internal edges as =E2=80=9Chypothetical=E2=80=9D=
However, under the paradigm of evolution, there is nothing more
hypothetical about these edges than there are about the named taxa
represented by specimens. If the inferred tree is correct, then these
ancestral taxa represented by these edges must have existed. Under the
evolutionary paradigm, the extent to which we treat named taxa (A, B, C)
as real entities of descent with modification is the extent to which we
treat internal lines as symbolizing real ancestors. They are not
=E2=80=9Chypothetical=E2=80=9D; they are simply unsampled and inferred (or,=
especially in systematics of fossil organisms, unrecognized or
misidentified as descendant species).
In Fig. 1B, we have added more information to the tree. Each numbered
black rectangle represents an evolutionary character hypothesized to be
fixed (sensu ) somewhere in the lineage represented by the edge to
which the rectangle is attached. (The placement of the rectangle within
an edge does not matter; for example, the tree in Fig. 1B does not
assert that apomorphies 3, 4, and 5 became fixed at different times just
because they are shown at different heights on the page. Moreover, one
cannot draw inferences about when characters originated; for example, it
is possible that character 2 originated in lineage z before character 1,
but went extinct in other lineages (such as A) and became fixed only in
the common ancestor y of B and C.)
Hennig () used the symbology of Gregg (), which Gregg apparently
derived from Woodger (). In a node-based tree, taxa are represented
by vertices, not by edges. An edge of a node-based tree does not
represent a lineage or anything else occurring in nature. Rather, an
edge simply represents a relationship among two vertices, or, in
phylogenetic parlance, the hypothesis of a relationship. Specifically,
an edge between a parent vertex X and a child vertex Y represents the
hypothesis that X is an ancestor of Y. This node-based tree
representation is fairly intuitive (at least to us) and likely how most
practicing evolutionary biologists interpret phylogenies.
Fig. 2: Modified version of Figure 14 of Hennig (1966, p. 59) entitled
=E2=80=9CThe species category in the time dimension.=E2=80=9D
Left: a stem-based tree. Letters are symbols for species and the number
applied to the letters are labels for samples of each species considered
at a particular time period. Right: a node-based tree with single-headed
arrows symbolizing relationship statements and circles representing
species. Note the correspondence between the lineages on the left and
the circles on the right, as shown by the brackets and double-headed
arrows for selected lineages and vertices.
Fig. 2 is redrawn from Hennig () and portrays the relationships
among samples of an evolving clade in two ways. The left-hand side of
Fig. 2 portrays a stem-based tree with lineages represented by edges
(species to Hennig) and sampled populations of these lineages placed in
time with circles (B1, B2, etc.). Vertices represent speciation events.
The right-hand side of Fig. 2 shows the node-based tree corresponding to
the stem-based tree on the left-hand side. Here the taxa are represented
by vertices (population samples being completely ignored). The edges
represent phylogenetic, not phenetic, relationships between these
species (i.e., genealogical relationships based on synapomorphies rather
than similarity relationships based on a metric or idea of overall
similarity). Hennig () makes this clear in a number of diagrams (see
his Figs. 4, 6, 14, 15) and in his text.
Equivalency of stem-based and node-based trees
Below, we prove mathematically that node-based trees and stem-based
trees carry the same information, albeit encoded in different ways. We
start by setting up some notation.
Let T be a tree with root vertex r. Recall that specifying a root for a
tree determines its =E2=80=9Cparent=E2=80=9D and =E2=80=9Cancestor=E2=80=9D=
relations completely. If x
is the parent of y, we will denote the edge joining them by the symbol x
=E2=86=92 y (in keeping with the convention that edges point from parents to
children). Alternately, we will write x > y to indicate that vertex x is
an ancestor of vertex y.
It is a standard fact that for every set X of vertices in T, there is a
unique vertex y (which may or may not belong to X ) with the following
two properties: first, y =E2=89=A5 x for every x in X, and second, if z is =
other vertex such that z =E2=89=A5 x for every x in X ,then z > y . The fir=
these conditions says that y is a common ancestor of the vertices in X;
the second condition says that it is the youngest common ancestor.
Finally, we call T a planted tree if its root r has only one child.
(=E2=80=9CPlanted=E2=80=9D is a more restrictive condition than =E2=80=9Cro=
oted=E2=80=9D; every planted
tree is necessarily rooted, but not vice versa.)
We now describe an equivalence between two different kinds of labeled
trees. Let n be any positive integer, and let T be a rooted tree with n
vertices, labeled 1, 2, =E2=80=A6, n . (Any of these may be the root of T.)
Construct a tree U from T according to the following algorithm.
Create a new root vertex, labeled 0, and create a new edge 0=E2=86=92r,
where r=3D root(T).
Label each edge v=E2=86=92w of this tree with the number w.
Erase the labels of the vertices.
An example of the construction of U from T is shown in Fig. 3. (The
vertex labels are shown in black, and the edge labels in red.) Note that
U has n +1 vertices, hence n edges, which are labeled 1, 2, =E2=80=A6, n . A
consequence of the construction is that U is always a planted tree,
because its root (from which the label 0 was erased) has exactly one
child, namely, r =3D root(T).
Fig. 3: The steps of Algorithm A, read A to D.
Reading D to A illustrates Algorithm B.
We can reconstruct T from U by reversing Algorithm A. Specifically,
suppose that U is any planted tree with n edges, labeled 1, 2, =E2=80=A6, n=
Note that U must have exactly n +1 vertices. Let r be the root vertex,
and let s be its unique child. Now, construct a tree T from U as follows:
Label each non-root vertex of U by the label of its parent edge, and
assign the label 0 to vertex r.
Erase all labels on the edges.
Delete vertex r and edge rs, and designate s as the root of the
These steps are exactly the reverse of those of Algorithm A; for an
illustration, see Fig. 3. It is worth mentioning that the algorithms
work the same way whether or not the input tree has polytomies (vertices
with more than two children). The algorithms establish the following
There is a one-to-one correspondence between the following two sets:
The set of all rooted trees T on n vertices labeled 1, 2, =E2=80=A6,n; =
The set of all planted trees U on n+1 vertices, with edges labeled
1, 2, =E2=80=A6,n.
Because the correspondence is one-to-one, the rooted tree T contains
exactly the same information as its planted counterpart U. However, one
must be careful when translating between T and U. For example, there is
not a one-to-one correspondence between arbitrary subtrees of T and
arbitrary subtrees of U. Indeed, if E is the set of edges of a subtree
of U, then the corresponding set of vertices of T will not form a
subtree unless E is planted. For example, suppose that T and U are as
shown in Fig. 3A and Fig. 3D, respectively. The edges 4, 5, 8, 9 form a
subtree of U, but vertices 4, 5, 8, 9 do not form a subtree of T; see
Fig. 4A, B. On the other hand, vertices 2, 4, 5, 8, 9 do form a subtree
of T because the corresponding edges 2, 4, 5, 8 and 9 form a planted
subtree of U; see Fig. 4C,D.
Fig. 4: In the node-based tree T (A), the vertices 4, 5, 8, 9 do not
form a subtree, even though edges 4, 5, 8, 9 form a subtree of the
corresponding stem-based tree U shown in B.
In the node-based tree T (A), the vertices 4, 5, 8, 9 do not form a
subtree, even though edges 4, 5, 8, 9 form a subtree of the
corresponding stem-based tree U shown in B. In contrast, the subtree of
T (C) formed by vertices 2, 4, 5, 8, 9, corresponds to the planted
subtree of U shown in D. The figure also illustrates possible
circumscriptions of the terminal taxa 4, 8, 9; heavy lines denote edges
included in the classification. Applying node-based circumscription to
the stem-based U results in the polyphyletic group of 4, 8, 9, and the
inferred ancestor 5; as shown in B, there is no edge connection to the
sister group comprising the terminals 6 and 7 because inferred ancestor
2 remains excluded (dashed line). In contrast, a node-based
circumscription of T or a stem-based circumscription of U (shown in C
and D) yields the monophyletic group composed of the terminal taxa 4, 8,
and 9 and their inferred ancestors 2 and 5.
Indeed, it follows from Algorithms A and B that there is a one-to-one
correspondence between proper subtrees of T and planted proper subtrees
of U. Similarly, there is a one-to-one correspondence between subtrees
of T (not necessarily proper) and planted subtrees of U (again, not
Additional biological information associated with a stem-based or
node-based tree can be translated via this algorithm. For instance, the
character data represented by edge labels in a stem-based tree (Fig. 1B)
can be represented by vertex labels in the corresponding node-based tree.
An example: node- and stem-based concepts of monophyly
While node-based and stem-based trees carry the same basic information
about taxa and ancestry, they represent this information in different
ways. Therefore, it should not be surprising that biological concepts
are modeled by different mathematical substructures in the two kinds of
trees. We provide an example of this through a discussion of how the
phylogenetic concept of monophyly is represented in each tree model.
Hennig=E2=80=99s (, pp. 206-209) discussion of monophyly admits only one
definition of this term; a monophyletic group is a group that includes
all descendants of a common ancestral species. Although not mentioned in
this section, elsewhere Hennig (:71) makes it clear that he intends
the ancestral species also to be a member of the group (and, indeed is
logically equivalent to all descendant members of the group). Recently,
additional means of circumscribing monophyletic groups were proposed
(   ), which have now been codified into formal rules
distinguishing several kinds of clade recognition. Two of these are
germane to our discussion.
Definition 1: =E2=80=9CA node-based clade is a clade originating with a
particular node on a phylogenetic tree, where the node represents a
lineage at the instant of a splitting event.=E2=80=9D (The PhyloCode versio=
January, 2010, Article 2.2, )
Definition 2. =E2=80=9CA branch-based clade is a clade originating with a
particular branch (internode) on a phylogenetic tree, where the branch
represents a lineage between two splitting events.=E2=80=9D ()
We argue that this distinction between node-based and branch-based (=3D
stem-based) concepts of monophyly arises from confusion between the two
types of trees we have discussed. This is not intended as a critique of
the entirety of the PhyloCode, but rather is provided as an example of
how being explicit regarding graphical models can provide clarity to
discussions of biological concepts. Indeed, given the discussion of
these tree models above and adopting Hennig=E2=80=99s (, p.71) usage of
=E2=80=9Cmonophyly=E2=80=9D, it is evident that a monophyletic group with c=
ancestor A is represented in a node-based tree T by the proper subtree
rooted at the vertex corresponding to A, and in a stem-based tree U by
the proper subtree planted at the edge corresponding to A. Recall that
the proper subtrees of T are in bijection with the planted proper
subtrees of U. To rephrase this observation, the correct mathematical
representation of monophyly can be found either by applying Definition 1
to a node-based tree, or by applying Definition 2 to a stem-based tree.
A node-based name cannot exist for a stem-based tree just as a
stem-based name cannot exist for a node-based tree. If it is agreed that
a tree must be either a node-based or a stem-based tree and not some mix
of the two, then one must select the appropriate naming scheme to
represent monophyly. While authors might argue for employing both
concepts of monophyly for a single phylogeny, they must then recognize
that such a phylogeny would not be a valid mathematical representation
of a tree.
It is worth examining what happens if we apply Definitions 1 and 2 to
the wrong kinds of trees. First, a =E2=80=9Cnode-based clade=E2=80=9D of a =
tree=E2=80=94speaking mathematically, a proper but non-planted subtree of a
stem-based tree=E2=80=94does not correspond to a monophyletic group of taxa.
Returning to the phylogenetic tree U shown in Fig. 3D, the non-planted
subtree highlighted in Fig. 4B is actually polyphyletic, not
monophyletic; every edge in U represents a taxon descended from taxon 2,
which does not belong to the subtree. That this set of taxa is
polyphyletic is perhaps clearer upon examining the corresponding
vertices in the node-based tree (see Fig. 4A.) This matches the
definition of =E2=80=9Ccrown clade=E2=80=9D. Second, a planted subtree in a=
tree (such as the tree spanned by the black vertices 1, 3, 6, 7 in Fig.
4A) is not monophyletic but paraphyletic, because it includes only one
child (3) of its root vertex while excluding child 2 and the children of
2). It is tempting to interpret such a tree as a stem-based clade that
includes a =E2=80=9Croot edge=E2=80=9D=E2=80=94here the edge from 1 to 3=E2=
=80=94but not its parent
vertex, here 1. However, the mathematical definition of a graph does not
permit such a structure; one cannot have an edge without both its
endpoints. Omitting the =E2=80=9Croot edge=E2=80=9D produces a well-defined=
contains the same biological information (regarded as a node-based
tree). If we are careful only to use the term =E2=80=9Cnode-based clade=E2=
working with node-based trees, and =E2=80=9Cstem-based clade=E2=80=9D when =
stem-based trees, then the two terms become synonymous. The difference
has no biological significance and lies only in the form of tree chosen
to represent the phylogeny. Both node-based and stem-based names as
proposed in the PhyloCode describe the single concept of monophyly,
albeit based on two possible tree graphs. Given that in empirical
phylogenetic studies all recognized monophyletic groups must be
corroborated by one or more synapomorphies (though not necessarily
unique and unreversed), we suggest that the PhyloCode be amended to
reflect this. A simple approach would be to state explicitly in the
PhyloCode one of the two graphical representations of trees for
reference and then apply the logical corresponding concept of monophyly
throughout the PhyloCode.
Practicing =E2=80=9Ctree-thinkers=E2=80=9D might easily make the mental con=
between node-based and stem-based trees. By explicitly detailing that
these tree models are mathematically equivalent, we aim to add clarity
to discussions related to the biological meaning of phylogenies. It is
important to be specific about these two distinct representations of
trees. During the latter half of the twentieth century, phylogenies
transitioned from being essentially cartoon-representations to graphical
representations of the results of an analysis of data (typically
represented in a matrix). We argue that biological concepts relating to
a phylogeny that is inferred based on an analysis of data should be
discussed in a context consistent with the graphical model used to
display results of the analysis. To our knowledge, most evolutionary
biologists do not construct estimates of phylogenetic relationships
based on mathematical models in which transformations of characters
occur at both nodes and along branches. Instead, computations are made
at either vertices (=3D nodes) or edges (=3D stems). We leave open the
possibility that authors might employ a workable mental model in which
character transformations occur along both nodes and branches, but we
argue that this would not be strictly representing the results of the
analysis. Last, and importantly, we add that representing relationships
between taxa via either a node-based or stem-based tree does not
preclude subsequent use of the same phylogeny to model processes that
might occur along both nodes and branches (as implemented, for example,
in the dispersal-extinction-cladogensis model of geographic range
evolution;  ). Without clear recognition of node-based and
stem-based trees, as well as the equivalency between these, authors may
arrive at confused interpretations of phylogenies, including
circumscriptions of monophyly.
The authors have declared that no competing interests exist.
EOW thanks the late David Hull (Northwestern University) for sending a
copy of a manuscript that he never published entitled =E2=80=9CHierarchies =
Hierarchies=E2=80=9D that touched upon the problems associated with
process/pattern and tree/cladogram controversies, and for what must have
seemed to him hours of discussion on things phylogenetic and
philosophical regarding the subject. We also thank Shannon DeVaney (Los
Angeles County Museum) and Mark Holder (University of Kansas) for
reading the manuscript and providing a critical review.
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