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their representative sequences). Suppose
D
=
D
T
,ω
for an
ω
edge-weighted phy-
logenetic
X
-tree
T
(with
n
). Then one can phrase the problem of finding a
(
n
edge-weighted) phylogenetic tree
T
that “best fits” the observed data
D
on
X
as one
of finding a pair
|
X
|=
is as small as possible. Equivalently,
one may take a
least-squares approach
, since minimizing
(
T
,ω)
∈
T
n
so that
δ(
D
,
D
T
,ω
)
δ(
D
,
T
T
,ω
)
is the same as
2
. When there is an exact fit,
minimizing
1
i
<
j
n
(
d
(
i
,
j
)
−
d
T
,ω
(
i
,
j
))
δ
=
0.
To summarize:
For a set
X
,
|
X
|=
n
, each positive edge-weighted phylogenetic
X
-tree
(
T
,ω)
.
For a fixed ordering of reading off upper diagonal elements in the matrix
D
T
,ω
one obtains a vector
givesrisetoatreemetric
D
T
,ω
n
2
.
m
m
v
D
,
T
,ω
∈
R
=
There are one-to-one correspondences between
-
pairs
∈
T
n
and positive edge-weightings,
-
the
n
by
n
dissimilarity maps
D
that are tree metrics
D
T
,ω
, and
-
the vectors
(
T
,ω)
∈
T
n
of trees
T
m
.
v
D
,
T
,ω
∈
R
m
, and match-
ing such an edge-weighted tree to a dissimilarity map
D
arising from evolutionary
distances on sequences corresponding to a set
X
of species (or genes, etc.) becomes
a problem of minimizing distances of the vectors
In this way, edge-weighted phylogenetic
X
-trees become points in
R
m
.
v
D
,
T
,ω
and
v
D
in
R
Regarding positively (or nonnegatively) edge-weighted trees
(
T
,ω)
with
T
∈
T
n
m
, the goal is, when presented with a point
D
m
,
as a subset of points
D
T
,ω
∈
R
∈
R
m
as is possible.
to seek one of these points
D
T
,ω
as close to
D
in
R
In this way, a fundamental problem of biology is turned into a geometric problem.
In particular, one wants to understand better the set
={
(
T
,ω)
|
T
=
(
V
,
E
)
∈
T
n
→
R
+
}
m
.Theset
T
n
,ω
:
E
regarded as a set of points
⊂
R
T
n
is often called
T
n
“tree space,” though there is one for each
n
.
Example 10.6.
3, then there is only one tree topology, and there is only
one (unrooted binary) phylogenetic tree
T
If
n
=
∈
T
3
, but infinitely many edge-weightings
ω
for any choice of
T
. More precisely, for each such
T
, there are three edges, and
for each edge
e
R
+
, independent of the other
,ω(
e
)
can take all positive real values
edges. The edge-weighting
ω
on
T
is completely described by the ordered triple
(ω(
e
), ω(
g
), ω(
h
))
. In this way, geometrically, the significance of a choice of edge-
3
of
R
+
3
weighting
ω
for
T
is that
ω
describes a point in the positive orthant
R
associated to
T
. Taking all possible positive weightings on
ω
on
T
corresponds pre-
R
+
3
=
∈
T
4
, there are
cisely to the full positive orthant
for
T
.For
n
4, for any
T
(
2
n
−
3
)
=
(
8
−
3
)
=
5 edges and
(
2
n
−
5
)
!! =
(
8
−
5
)
!! =
3 phylogenetic
X
-trees
R
+
5
. (For those not
so familiar with high-dimensional geometry, this is analogous to the way in which,
for every point in time, there is a three-dimensional copy of space at that point in time,
only here the finitely many trees
T
T
∈
T
4
. The pairs
(
T
,ω)
correspond to points in three copies of
∈
T
n
play the role of selecting out just finitely
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