Editing Neighbor joining (section)

=== Second step ===
====Second joining====
The corresponding <math>Q_2</math> matrix is:

{| class="wikitable" style="text-align: center"
|-
! style="width: 20px;" | 
! style="width: 20px;" | u
! style="width: 20px;" | c
! style="width: 20px;" | d
! style="width: 20px;" | e
|-
! u
| 
| style=background:#ffffcc; | &minus;28
| &minus;24
| &minus;24
|-
! c
| style=background:#ffffcc; | &minus;28
| 
| &minus;24
| &minus;24
|-
! d
| &minus;24
| &minus;24
| 
| style=background:#ffffcc; | &minus;28
|-
! e
| &minus;24
| &minus;24
| style=background:#ffffcc; | &minus;28
| 
|-
|}

We may choose either to join <math>u</math> and <math>c</math>, or to join <math>d</math> and <math>e</math>; both pairs have the minimal <math>Q_2</math> value of <math>-28</math>, and either choice leads to the same result. For concreteness, let us join <math>u</math> and <math>c</math> and call the new node <math>v</math>.

====Second branch length estimation====
The lengths of the branches joining  <math>u</math> and <math>c</math> to <math>v</math> can be calculated:

:<math>\delta(u,v)=\frac{1}{2}d(u,c)+\frac{1}{2(4-2)} \left [ \sum_{k=1}^4 d(u,k) - \sum_{k=1}^4 d(c,k) \right ] \quad =\frac{7}{2} + \frac{20-22}{4} = 3</math>
:<math>\delta(c,v)=d(u,c)-\delta(u,v) \quad = 7-3 = 4</math>

The joining of the elements and the branch length calculation help drawing the neighbor joining tree [[#NJ_with_5_taxa|as shown in the figure]].

====Second distance matrix update====
The updated distance matrix <math>D_2</math> for the remaining 3 nodes, <math>v</math>, <math>d</math>, and <math>e</math>, is now computed:

:<math>d(v,d)=\frac{1}{2} [d(u,d)+d(c,d)-d(u,c)] = \frac{7+8-7}{2} = 4</math>
:<math>d(v,e)=\frac{1}{2} [d(u,e)+d(c,e)-d(u,c)] = \frac{6+7-7}{2} = 3</math>

{| class="wikitable" style="text-align: center"
|-
! style="width: 20px;" | 
! style="width: 20px;" | v
! style="width: 20px;" | d
! style="width: 20px;" | e
|-
! v
| 0
| '''4'''
| '''3'''
|-
! d
| '''4'''
| 0
| ''3''
|-
! e
| '''3'''
| ''3''
| 0
|-
|}