Phylogenetic Tree Reconstruction: Geometric Approaches - Mathematical Concepts and Methods in Modern Biology - page 311

FIGURE 10.2
Quartet ((,),(,));

Exercise 10.6.

1. (Quartets) There are three distinct unrooted binary phylogenetic X -trees on a set

of four leaves X . That is, up to isomorphisms of binary phylogenetic X -trees, there
are only three distinct such non-equivalent trees—every other unrooted binary
phylogenetic X -tree on four leaves X will be an unrooted binary

phylogenetic X -tree equivalent to one of these. Draw (representatives of) these
three trees.
2. Up to equivalence, how many distinct rooted binary phylogenetic X -trees on a
fixed set X of three leaves are there? Draw representatives of each.
3. For X

={ 1 , 2 , 3 , 4 }

, how many distinct tree topologies are there underlying the
rooted phylogenetic X -trees? What if these X -trees are unrooted?

={ 1 , 2 , 3 }

More formally, there is an equivalence relation on the set of all rooted phylogenetic
trees (resp., unrooted phylogenetic trees) given by setting two such trees to be related if
they are equivalent phylogenetic X -trees. Just as with equivalence relations in general,
one often picks a representative of each equivalence class and identifies thewhole class
with any representative (e.g., just as a fraction in lowest common terms represents all
fractions which reduce to it). Consequently, from now on, for any fixed X , when we
speak of the set of “all rooted phylogenetic X -trees” or “all unrooted phylogenetic
X -trees” we understand that any particular such tree can be represented by one on
X with the same underlying tree topology. For (rooted) phylogenetic X -trees,

this formalizes the notion in biology of a “cladogram,” so when drawn in the plane,
neither horizontal axis nor vertical axis has any particular meaning, e.g., as in [ 4 ,p.15].
From now on, for any set X of n elements, we will let

=[ n ]

T n denote the set of all
unrooted binary phylogenetic X-trees T. For example, by Exercise 10.4 , T 4 consists

of three distinct such trees (again, up to equivalence), and by Exercise 10.5 , all these
leaf-labeled trees in T 4 have the same underlying tree topology.

Exercise 10.7.

a. Identify the subsets of trees below which are isomorphic to one another simply
as trees (see Figure 10.3 ).
b. For the same collection of trees, identify which are isomorphic as phylogenetic
X -trees, for X =[ 6 ] .