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is a phylogenetic
X
-tree and
T
=
(
V
,
E
)
is a phylogenetic
X
-tree. Since
T
and
T
are trees with additional properties, and since isomorphisms of
directed trees preserve indegrees and outdegrees, respectfully, preserves degrees for
undirected trees, a function
Suppose
T
=
(
V
,
E
)
T
can be an isomorphism of the phylogenetic
ψ
:
T
→
trees
X
and
X
only if
X
on the leaf sets.
ψ
restricts to a bijection
ψ
:
X
→
X
|
T
Necessarily, then,
|
X
|=|
. A tree isomorphism
ϕ
:
T
→
for which the
X
of
V
is in fact an identity map (so
X
X
and
restriction
ϕ
:
X
→
ϕ
:
V
→
=
ϕ(v)
=
v
X
), is, by definition, an isomorphism of phylogenetic
X
-trees. In
our biological context, the notion of isomorphismmakes explicit those ways in which
pictures of phylogenetic trees might differ, but still represent the same evolutionary
relationships among the leaves.
Example 10.3.
for all
v
∈
,let
T
1
be the unrooted binary phylo-
genetic
X
-tree
((A,B),(C,D));
where the cherry
For
X
={
A
,
B
,
C
,
D
}
{
A
,
B
}
has ancestor
u
and
the cherry
has ancestor
v
.Let
T
2
be the unrooted binary phylogenetic tree
((A,C),(B,D));
where cherry
{
C
,
D
}
{
A
,
C
}
has ancestor
s
and cherry
{
B
,
D
}
has ances-
tor
t
. Then setting
ϕ
:
T
1
→
T
2
given by
ϕ(
A
)
=
C
,ϕ(
B
)
=
D
,ϕ(
u
)
=
s
,ϕ(v)
=
t
B
creates an isomorphism of
T
1
with
T
2
as phyloge-
netic
X
-trees. (Note that the lengths of edges play no part here, only the connectivity
relationships.)
Exercise 10.5.
,ϕ(
C
)
=
A
, and
ϕ(
D
)
=
1.
For
X
,let
T
1
be the unrooted binary phylogenetic
X
-tree
((A,B),
(C,D));
and let
T
2
be the unrooted binary phylogenetic tree
((A,C),(B,D));
and let
={
A
,
B
,
C
,
D
}
T
2
be as in as in Example
10.3
.
a.
Explain why although
ϕ
:
T
1
→
T
2
is an isomorphism of trees, it is not an
isomorphism of phylogenetic
X
-trees.
b.
Find another distinct isomorphism of
T
1
and
T
2
just as trees.
2.
Explain why trees
(A,(B,C));
and
(A,(C,B));
are isomorphic as phylo-
genetic
X
ϕ
:
T
1
→
trees, but
(A,(B,C));
and
(B,(A,C));
are not.
3.
Explain why if
(A,(B,C));
and
(2,(1,3));
are isomorphic as phyloge-
netic trees, then
(A,(B,C));
and
(1,(2,3));
are not.
={
A
,
B
,
C
}
Formally, then, the
unlabeled
or
underlying
tree associated to a phylogenetic
X
-tree
T
is an equivalence class of all trees
T
isomorphic to
T
. Informally, it is represented
by any tree
T
isomorphic to
T
, and
T
is said to have the same
topology
or
tree
topology
as
T
.
Example 10.4.
There is only one tree topology for unrooted phylogenetic
X
-trees on
aset
X
of four leaves; it is given in Newick form by
((,),(,));
(see Figure
10.2
).
If there is an isomorphism
T
of rooted phylogenetic
X
-trees, then
T
and
T
are said to be
equivalent trees
, and likewise for two unrooted phylogenetic
X
-trees.
ϕ
:
→
T
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