Biology Reference
In-Depth Information
Example 10.2.
1.
The trees in parts (1)-(5) of Exercise
10.1
were phylogenetic
X
-trees with the
leaves labeled by
X
, whereas the interior vertices were not labeled.
The tree in part (8) of Exercise
10.1
had no labeled leaves and hence is not a
phylogenetic
X
-tree, but does represent the “underlying tree” or “tree topology”
of an unrooted phylogenetic
X
-tree on three leaves, as will be defined a little later.
2.
An approximation to the evolutionary history of all organisms alive today would
be a phylogenetic
X
-tree wherein the leaves
X
consist of all species currently in
existence; in this case,
={
A
,
B
,
C
}
000. The Tree of Life project
http://
tolweb.org/tree/
,
while noting that a tree is an imperfect model, uses phy-
logenetic
X
-trees as an organizational framework, and the use there is represen-
tative of hundreds of more specialized scientific articles employing phylogenetic
trees. The reader is encouraged to explore this website for current versions and
many visualizations of phylogenetic trees on multiple organizational levels of
species and other biological taxa.
|
X
|
>
1
,
800
,
A tree may simply be called
phylogenetic
if it is a phylogenetic
X
-tree for some
(finite) set
X
. Without loss of generality, it is possible to use
[
n
]={
1
,
2
,...,
n
}
in
place of
X
for
n
. A binary tree that is a phylogenetic tree is called a
binary
phylogenetic tree
. Binary phylogenetic
X
-trees form the most commonly used group
of trees in biology, since branching events that arise from evolutionary events, like
mutations or speciation, correspond to a splitting of one lineage into two, rather than
more than two simultaneously. However, non-binary trees do arise when ancestry is
uncertain (e.g., the order of speciation or other branching events is uncertain), and in
this case, there may be sisters that are not just cherries. An extreme example is a
star
tree
on
n
leaves, which has
n
|
X
|=
+
1 nodes and
n
edges, with the single interior vertex
joined to each and every leaf.
Exercise 10.4.
1.
Show any rooted binary tree on
n
3 leaves always has a cherry, and hence
3 leaves does, too.
2.
Explain why the total number of vertices of any unrooted binary tree on
n
conclude any unrooted binary tree on
n
3
2.
3.
Argue that the number of edges of any unrooted binary tree on
n
leaves is 2
n
−
3leavesis
3.
4.
Suppose
n
2
n
−
3. Using the previous part of this exercise, argue that the number of
unrooted binary phylogenetic
X
-trees for a set
X
of
n
leaves is the double factorial
(
. As a hint, consider how to build an unrooted binary phylogenetic
X
-tree
on
n
leaves from any one on
n
2
n
−
5
)
!!
1 leaves by adding a new edge ending in the new
leaf to an edge on the tree with one less leaf. (By definition of the double factorial,
1
−
!! =
,
!! =
·
=
,
!! =
·
!! =
·
·
=
,
!! =
·
!! =
·
·
·
=
1
3
3
1
3
5
5
3
5
3
1
15
7
7
5
7
5
3
1
105,
(
−
)
!! =
(
−
)
·
(
(
−
)
−
)
!! =
(
−
)
·
(
−
)
···
·
·
and, generally,
2
n
5
2
n
5
2
n
1
5
2
n
5
2
n
7
5
3
1.
(
2
n
−
4
)
!
(Alternatively, one can show that
(
2
n
−
5
)
!! =
2
n
−
2
,for
n
2.)
(
n
−
2
)
!
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