Editing Distance matrix (section)

===== Additive tree reconstruction =====
Additive tree reconstruction is based on additive and ultrametric distance matrices. These matrices have a special characteristic:

Consider an additive matrix {{Math|M}}. For any three species {{Math|i, j, k,}} the corresponding tree is unique.<ref name=":0" /> Every ultrametric distance matrix is an additive matrix. We can observe this property for the tree below, which consists on the species {{Math|i, j, k}}.
[[File:Unique_tree_additive_matrix.png|center|frameless|Phylogenetic tree from 3 species]]
The additive tree reconstruction technique starts with this tree. And then adds one more species each time, based on the distance matrix combined with the property mentioned above. For example, consider an additive matrix {{Math|M}} and 5 species {{Math|''a'', ''b'', ''c'', ''d''}} and {{Math|''e''}}. First we form an additive tree for two species {{Math|''a''}} and {{Math|''b''}}. Then we chose a third one, let's say {{Math|''c''}} and attach it to a point {{Math|''x''}} on the edge between {{Math|''a''}} and {{Math|''b''}}. The edge weights are computed with the property above. Next we add the fourth species {{Math|''d''}} to any of the edges. If we apply the property then we identify that {{Math|''d''}} should be attached to only one specific edge. Finally, we add {{Math|''e''}} following the same procedure as before.