https://people.math.wisc.edu/~roch/research_files/review-steel-ams.pdf

Phylogeny—discrete and random processes in evolution1 by Mike Steel,
CBMSNSF Regional Conference Series in Applied Mathematics, 89, Society
for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
xvi+293 pp.
ISBN: 978-1-611974-47-8, List Price $64.00, SIAM Member Price $44.80, Order
Code: CB89

GREAT PDF

On Monday, April 3, 2023 at 6:11:09 AM UTC-4, Popping Mad wrote:
> https://people.math.wisc.edu/~roch/research_files/review-steel-ams.pdf
>
>
> Phylogeny—discrete and random processes in evolution1 by Mike Steel,
> CBMSNSF Regional Conference Series in Applied Mathematics, 89, Society
> for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
> xvi+293 pp.
> ISBN: 978-1-611974-47-8, List Price $64.00, SIAM Member Price $44.80, Order
> Code: CB89
>
>
>
> GREAT PDF

The first section really appeals to me as a mathematician. It uses standard graph theory terminology.
It's nice to see that there is a word for the only kind of tree that Harshman deals with:
"binary," defined with mathematical precision as a tree in which each interior vertex (i.e., not a leaf) has degree 3.
[I'll explain in a later post to this thread why "binary" is used instead of "ternary."]

Biologically speaking, one very important concept defined in the first section is "phylogenetic diversity."

"A natural goal might be to pick a set of species to protect that is “as diverse as possible” from an evolutionary point of view. But how to define phylogenetic diversity precisely?"

A better word than "diversity" would be "disparity," due to the rather counterintuitive meaning
given to "diversity" as "number of species". But let that pass.

The article provides a mathematical modeling of phylogenetic diversity, using weights attached
to each edge of the tree:

"The edges of a phylogenetic tree T are often associated with weights {we}e∈E(T) which may represent either the time elapsed or the expected amount of evolution (e.g., in number of mutations in a segment of the genome) along that edge."

For this first post, I am skipping over how phylogenetic diversity is calculated,
and moving to an issue that has caused untold confusion down through the decades,
both in s.b.p. and in the big outside world: how closely related are two species?

This is quantified in the article by using the best weights we can assign to each edge in the tree,
and adding together the ones on the unique path through the tree from species A to species B.
This is called "the path metric" between A and B.

The article avoids defining "related" [specifically, "more closely related"]
because of the way the systematists who dominate systematics define it.
Their method cannot be quantified because of its rudimentary nature.
And it is totally at odds with the quantification I gave above.

The dominant definition is the analogue of saying,
"Mitochondrial Eve is more closely related to everyone on earth today than she was
to anyone alive before she had children, including her parents and her siblings, if any."

On the other hand, look at the sentence "Species A is more closely related to species B
than it is to species C because the path metric from A to C is greater than the one from A to B."
This corresponds closely to the way we, including genealogists, use "more closely related" in everyday life.

Peter Nyikos
Professor, Dept. of Mathematics -- standard disclaimer--
University of South Carolina
http://people.math.sc.edu/nyikos