Different representations of dendrograms

I have a dendrogram represented in a format I don't understand:

(K_5:1.000030e+00,((K_1:2.000000e-05,(K_2:1.000000e-05,K_3:1.000000e-05):1.000000e-05):1.000000e-05,K_4:3.000000e-05)0.806:1.000000e+00):0.000000e+00;

I am not sure how to interpret the above. It is an output of hierarchical clustering.

K_1, K_2, K_3, K_4, K_5 are the data points.

I have other dendrograms represented in the following format:

[x_1,x_2,x_3,x_4,x_5] (we start with one big cluster and split a cluster at each step)

[x_1,x_2][x_3,x_4,x_5]

[x_1,x_2][x_3,x_5][x_4]

[x_1][x_2][x_3,x_5][x_4]

[x_1][x_2][x_3][x_5][x_4]

I want a way to convert between these two representations.

• Can you tell from which package you got this dendrogram output? Aug 3 '20 at 13:29
• It's from FINEstructure. It's for genome analysis. Aug 3 '20 at 14:17

This output represents the dendogram as a tree. The innermost parentheses represent the deepest parts of the tree. For instance the top (root) of the tree start with the pair K5 and a subtree, then this subtree is made of another subtree and K4, and so on.

If we ignore the numerical values (distances I assume?) we have this:

(K_5,
(
(K_1,
(
K_2,K_3
)

)
K_4
)
)

Which represents this tree:

--------------------
|                  |
|            -------------
K_5           |           |
-------       K_4
|     |
K_1  -----
|   |
K_2 K_3

Then it can be converted to the desired format:

[K_1 , K_2 , K_3 , K_4 , K_5]
[K_1 , K_2 , K_3 , K_4] [K_5]
[K_1 , K_2 , K_3] [K_4] [K_5]
[K_1] [K_2 , K_3] [K_4] [K_5]
[K_1] [K_2] [K_3] [K_4] [K_5]
• Is there a name for this representation? Is it the most standard representation? Aug 3 '20 at 15:10
• I don't know any name or whether this is a standard representation of a dendogram. However it's common to represent any kind of tree with a parenthesis structure, for instance mult(6,plus(3,4)) for "6 * (3+4)). I guess the notation is inspired from that... or it could simply be the most direct way to print the data structure in the software you're using. Aug 3 '20 at 15:19