Page 40 - Tequio 11

38 On some polytopes in phylogenetics/Hoessly/27-40

Figure 9.
The binary trees on X = fx1; x2; x3; x4g

∈ ℝ with coordinates lexicographic order have the form
±
The corresponding vectors ™ 2
( -* , -A , -ë , *A , *ë , Aë ) and are given by

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
= ( , , , , , ).
™ . = ( , , , , , ), ™ . = ( , , , , , ), ™ .
2 4 4 4 4 2 4 2 4 4 2 4 4 4 2 2 4 4

Figure 10.
The BME(4) polytope is given by a triangle in ℝ
6

5
The BME(n) polytope / ⊆ ℝ / has dimension ® © − , as there are exactly linear independent equations
14
/
® ©
*
obeyed by / (Eickmeyer et al., 2008). Furthermore it has (2 − 5)!! vertices, which is | / |, the cardinality of
the set / (see, e.g., (Semple & Steel, 2003)). Some known results are summarized in the following table.

14 A set of vectors is linearly independent if none of the vectors in the set can be defined as a linear combination of the others.
14
A set of vectors is linearly independent if none of the vectors in the set can be defined as a linear combination of the others.

Tequio, enero-abril 2021, vol. 4, no. 11