Cladistics

Consider, again, these taxa, characters and states:

internal atrial two temporal pedicillate
Taxon amnion legs scales blood nostrils septum fenestrations hemipenes gizzard teeth feathers wings vertebrae
perch no no yes cold no no no no no no no no yes
coelocanth no no yes cold yes yes no no no no no no yes
salamander no yes no cold yes yes no no no yes no no yes
frog no yes no cold yes yes no no no yes no no yes
turtle yes yes yes cold yes yes no no no no no no yes
man yes yes no warm yes yes no no no no no no yes
gecko yes yes yes cold yes yes yes yes no no no no yes
snake yes yes yes cold yes yes yes yes no no no no yes
alligator yes yes yes cold yes yes yes no yes no no no yes
budgy yes yes no warm yes yes yes no yes no yes yes yes

The preceding matrix, again, can be represented numerically (for convenience) as:

1 2 3 4 5 6 7 8 9 10 11 12 13
perch 0 0 0 0 0 0 0 0 0 0 0 0 0
coelocanth 0 0 0 0 1 1 0 0 0 0 0 0 0
salamander 0 1 1 0 1 1 0 0 0 1 0 0 0
frog 0 1 1 0 1 1 0 0 0 1 0 0 0
turtle 1 1 0 0 1 1 0 0 0 0 0 0 0
man 1 1 1 1 1 1 0 0 0 0 0 0 0
gecko 1 1 0 0 1 1 1 1 0 0 0 0 0
snake 1 1 0 0 1 1 1 1 0 0 0 0 0
alligator 1 1 0 0 1 1 1 0 1 0 0 0 0
budgy 1 1 1 1 1 1 1 0 1 0 1 1 0

Hennigian argumentation proceeds as follows: Each apomorphic (derived) character state defines a relationship. That is, the presence of an amnion defines the group (turtle, man, gecko, snake, alligator, budgy). However, the absence of an amnion does NOT define the group (perch, coelocanth, salamander, frog) as having descended from a common ancestor exclusive of turtle, man, gecko, snake, alligator, and budgy. That is,
character 1 defines the group:
but not the group:

We can then enumerate for all apomorphic states, for all characters, the group(s) hypothesized by each:

Having determined each group defined by each character, you can then begin, stepwise, to resolve the tree by adding the charatacters one at a time...

This is only really possible for relatively uncomplicated datasets of relatively small sizes. I there are lots of taxa and there is lots of homoplasy, it is not likely [sic] that you'll actually get the shortest possible tree or all of the shortest trees even if you happen to get one of them.

The Wagner algorithm provides a better way to arrive at an efficient tree. It is not guaranteed to be the shortest tree for complicated data, but it is guaranteed to get the shortest tree if there is no homoplasy, and is likely [sic] to do a lot better than you can by-eye.

We begin by constructing a Manhattan matrix of absolute character differences as follows...

The difference for any two taxa is simply the number of characters that are different for the two taxa:

	perch	coelocanth	salamander	frog	turtle	man	gecko	snake	alligator	budgy
perch		2	5	5	4	6	6	6	6	3
coelocanth			3	3	2	4	4	4	4	8
salamander				0	3	3	6	6	5	7
frog					3	3	6	6	5	7
turtle						2	2	2	2	6
man							4	4	4	4
gecko								0	2	6
snake									2	6
alligator										4
budgy

Now we can pick any two taxa to join, but normally this will be the two closest taxa, so, in this case we can choose either salamander-frog or snake-gecko.

Next we need to calculate the patristic distance between each of the yet-to-be-joined taxa to the existing interval INT(salamander:frog). This is simply, the sum of the patristic distances of the taxon-to-be-joined, less the patristic distance of the interval itself, all divided by two. For example,

D[perch: INT(frog:salamander)] = |(perch:salamander + perch:frog - frog:salamander)|/2
or...

= |(5 + 5 -0)|/2

= 5

Thus, our new matrix looks like this:

perch coelocanth turtle man gecko snake alligator budgy Int[SalFrog]
perch 2 4 6 6 6 6 3 5
coelocanth 2 4 4 4 4 8 3
turtle 2 2 2 2 6 3
man 4 4 4 4 3
gecko 0 2 6 6
snake 2 6 6
alligator 4 5
budgy 7

Thus, there are three possible choices for the next taxon to add to point X on the interval of salamnader:frog. Ideally, you'd want to persue all possible paths (e.g., mhennig in Hennig86), but here we'll just go with the first one that appears in the matrix (e.g., hennig in Hennig86) and keep going.

Now we need to find the shortest patristic distance among all of the possible additions of taxa to each of the three intervals

salamader:X

X:frog

X:coelocanth
but to do this we need to get the state-set for X. This is merely the median state for the three closest nodes, in this case,

coelocanth	0	0	0	0	1	1	0	0	0	0	0	0	0
salamander	0	1	1	0	1	1	0	0	0	1	0	0	0
frog	0	1	1	0	1	1	0	0	0	1	0	0	0
.
X	0	1	1	0	1	1	0	0	0	1	0	0	0

For perch,

D(Int salamander:X) = 5+5-0/2 = 5

D(Int frog:X) = 5+5-0/2 = 5

D(Int coelocanth:X) = 2+5-3/2 = 4
For turtle,

D(Int salamander:X) = 3+3-0/2 = 3

D(Int frog:X) =3+3-0/2 = 3

D(Int coelocanth:X) = 2+3-3/2 = 1
For man,

D(Int salamander:X) = 3+3-0/2 = 3

D(Int frog:X) =3+3-0/2 = 3

D(Int coelocanth:X) = 4+3-3/2 = 2
and so on... , in this case, D [turtle: Int(coelocanth:X)] = 1 is among a few possible shortest next-additions (1 step), so we can add turtle a follows:

Continue this process stepwise and you have calculated a wagner tree by stepwise addition, which might look something like this: