To calculate the retention index for a phylogenetic tree, we use the following formula:

$$\frac{\text{maximum number of steps on tree - number of steps on the tree}} {\text{maximum number of steps on tree - minimum number of steps in the data}}$$

To calculate the maximum number of steps, we look at the number of observed character states for a character and take the lowest number. For each character, we take the lowest number of observed traits and then sum across all characters.

Take the following dataset (columns are characters, taxa are rows):

Outgroup  000000
Taxa A    100001
Taxa B    110001
Taxa C    111001
Taxa D    111101
Taxa E    111111
Taxa F    111110

The maximum number of steps for the first character is 1, since the character state 0 is attested once and character state 1 occurs six times.

What I don't understand is why we do not use 6 for the maximum number of changes for the first character.

  • $\begingroup$ Could you explain what is the matrix? Each column is a feature, but what does 0 or 1 represent, a step? Perhaps I would understand better if you could show the retention index of this dataset $\endgroup$
    – llrs
    Commented Oct 25, 2018 at 7:57
  • $\begingroup$ The retention index is used to gain insight into how well the data conform to the most parsimonious tree. 0 and 1 represent the character state. $\endgroup$ Commented Oct 25, 2018 at 14:37
  • $\begingroup$ Agreed RI is used for a parsimony tree. The minimum of steps requires searching the total tree space of the data and is not trivial to obtain without an algorithm. The maximum number of steps would computationally derrived. $\endgroup$
    – M__
    Commented Dec 28, 2018 at 21:16
  • $\begingroup$ found this very useful. I was wondering you could help @Kohl, how do I go about working out max steps if I have a matrix with not only 0s and 1s, but other integers such as: eg taxonA 11000203044. <br/> taxonB 11001102034 Thanks for your help, hope you see this after it being closed for so long $\endgroup$
    – user8918
    Commented May 26, 2020 at 10:28

1 Answer 1


The maximum number of steps (or changes) is the number of taxa with state 1 or 0, whichever is smaller. Conceptually, this is the number of character changes if each taxon evolved its state independently of the other taxa.

If for a given character the character state 0 occurred twice and character state 1 occurred five times, 2 would be the maximum number of changes.

  • $\begingroup$ Thank you for this. Could you just edit your answer to explain why in the scenario you sketch above the five-time occurrence of character state 1 is not used in calculating the maximum number of steps? What I find confusing is that if character state 1 occurs five times, could that not represent five independent (i.e., homoplasious) character state changes? $\endgroup$
    – Namenlos
    Commented Oct 25, 2018 at 15:54
  • $\begingroup$ "maximum number of steps" corresponds to the maximum number of steps that can be reconstructed using Fitch parsimony. Five changes from 0→1 is possible, but less parsimonious than two changes from 1→0. Less parsimonious still, but still compatible with the data, is a million unobserved steps from 0→1 and back again. $\endgroup$
    – ms609
    Commented Apr 2 at 9:54

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