Stopping Rules for Two-Sigma Structural Monophyly in Morphology-Based High-Resolution Phylogenetics

Structural monophyly is a model of how ancestral species and their descendants are interrelated, given the fact that about half of extant species are progenitors of one or more descendant species [1,2]. Structural monophyly knits together globally shared evolutionary processes that create stable ecosystems well adapted to present circumstances and have banked effective traits that provided survival in past, episodic environmental changes. Both the minimum and maximum Shannon entropies [3,4] are involved [5]. Entropy [6,7] is minimized by the redundancy involved as a progenitor species shares its newest traits (the novon set) with all its immediate descendant species.

In minimally monophyletic groups, there are usually four new traits shared among the single progenitor and a few descendant species (that trait set is the immediate ancestron). Entropy is maximized when each descendant species generates novel trait changes that are lost to the lineage only at the very base of a caulogram (a stem-taxon evolutionary tree). These traits exist as expressed traits somewhere in the lineage, and also as different states of the same character, being then potential traits in the lineage’s character set (the ancestron). Minimized entropy at the genus level and maximized entropy at the supra-generic level (tribe, subfamily, family) combine to enhance diversity in species in the present, and survival of the lineage through geologic time.

The two processes of redundancy and preservation of traits are revealed only with direct analysis of expressed traits (here, “morphology” includes all expressed traits). The techniques of high-resolution phylogenetics in two-sigma structural monophyly are explained below.

A fundamental criterion for completing analysis in science is the stopping rule [8,9]: “When does one stop sampling and analyzing?”. The prima facie answer is: “When you have the publishable results”, that is, results that are meaningful in the context of the larger research field. There are two ways to implement a stopping rule. First, pre-decide on a result: ask a testable question and estimate a reasonable answer with sufficient support against alternative answers. Second is to continue sampling and analyzing to narrow down the results until there is essentially no change in the results and the results are meaningful in the larger context. Commonly, stopping rules involve probabilistic measures, as in physics, where five-sigma may be required for exclusion of alternative explanations. In the analysis of structural monophyly, a 95% credible interval is considered sufficient for strong hypotheses of evolutionary relationship.

This paper reviews and consolidates published research. What is new is the clarification of stopping rules and the idea that the exclusion of two sigma of uncertainty is a major feature of taxon individuation.

The sticking points are the degree of expectedness or surprise in the results, and the congruence with causally related processes, plus the overall completeness and degree of statistical confidence in the results [10]. This paper presents high-resolution morphological analysis of structural monophyly in the context of molecularly integrated, total data evaluation of evolutionary processes as descent with modification. The basic stopping rule is a statistically well-supported ancestor-descendant evolutionary tree based on total evidence. This is as opposed to the now common low-resolution coalescent Bayesian or maximum likelihood generation of molecular clades [11,12] with mapping of morphology to molecular cladograms as an investigation of common ancestry. The stopping rule in the latter case is a statistically well-supported cladogram of shared ancestry, largely based on molecular evidence from spacers, introns, or, more recently, exons.

A familiar example of a stopping rule used in standard taxonomic revisionary work is where one studies expected variation and distinctions between species, with a view to producing nomenclatural distinctions of correct names and homo- and heterotypic synonymy, descriptions, geographic distributions and lists of specimens examined, and dichotomous keys to species, paralleling research in other groups. To do this, one samples nature (including museum specimens) until there is little added to observed variation with continued sampling, samples the literature until little more is learned with additional reading, writes descriptions that are parallel to other published descriptions (or perhaps adding new traits) to aid in comparison, and generates keys to species based on those two or more traits that correlate well. The stopping rule is largely “is it publishable” in being parallel to or a build on previous work. It should be relatable to similar groups and reflect similar theoretical evolutionary processes. The stopping rule is generally a prior estimate of the expected result, and sequential sampling narrows the study in that direction.

For extension of taxonomic information into the realm of evolutionary analysis, additional stopping criteria are both valid and necessary. High-resolution phylogenetics modeling structural monophyly [1] uses more complete stopping rules than does cladistics-based phylogenetics, which stops analytic study far too soon, as will be explained. Protocols are given here for each step in the evaluation of structural monophyly.

This paper presents a broad review of the changing relationships of progenitor and descendant species on an evolutionary tree. The technique is fundamentally based on standard taxonomic methods used to model evolutionary trees as descent with modification. These trees are called caulograms because nodes at branches are usually identifiable as actual species (if not, then their nodal trait changes can be inferred). This is as opposed to more common cladistic models generating cladograms of common ancestry [13,14]. Descent with modification identifies extant (or rarely inferred) ancestral species of extant descendant species and demonstrates how character states have changed during speciation. Morphological traits are used, being directly associated with adaptation and differential survival. Examples are presented with a comparison with cladistics.

Aspects of the methodology for determining structural monophyly in a taxonomic group are discussed, explained, and reviewed for presently accepted stopping rules. Papers examined are those of the present author, the only researcher to date to have detailed minimally monophyletic groups as such. These methods are compared to the methods of cladistic phylogenetics. The calculations for doing the Shannon-Turing sequential Bayesian analysis are simple and intuitive and are detailed in nearly all the papers by the authors cited in the present paper. Mathematical calculations were done with Desmos graphing software (version 7.37.0.0) and Microsoft Excel spreadsheet.

This summary of structural monophyly methods is needed because the techniques have been developed over many years and published in scattered journals. Analysis by descent with modification is now sufficiently mature that a single paper can bring together the fundamental concepts required for adequate analysis. Explanatory and exemplifying illustrations are largely taken from the author’s previously published papers [1] with some modification. This is a “how-to” presentation designed to ease taxonomists and evolutionists into a simplified analysis that avoids the problems of cladistic-based practices, which are detailed separately below.

The basic unit of evolution is the species, defined here as any species concept that lends itself to the methods determining structural monophyly. The basic unit of evolutionary process, however, is the genus, which has a definite and predictable structure that controls the extent and substance of speciation. The genus is here defined as a minimally monophyletic group of one progenitor species and usually up to four (sometimes more) descendant species, each of which shares the newly evolved traits of the progenitor.

In structural monophyly analysis, a character state is termed a “trait”, a definitive descriptor of one apparent adaptive novelty of a newly evolved species. There are usually about four traits that are newly generated during speciation by character state changes in the ancestral species character set.

It has long been known [15] that the numbers of species in a genus, and genera in a family, when graphed produce an inverse function, that is, a hollow curve with high values on the y-axis becoming asymptotically lower on the x-axis with increasing x values. Examples are given below (Figure 1). Dozens of similar hyper-geometrical graphs are given for different taxonomic groups by Willis [15] in his Age and Area treatise, all implying that in past taxonomic studies most genus have between 1 and 5 species, with numbers of genera with more species being increasingly much fewer. The superabundance of examples of many groups with inverse function graphs of species per genus provides a stopping point. Exemplary graphs in Figure 2 are provided by Stevens [16] for vascular plant species per genus, and by François [17] for reptile species per genus. Demonstration of the scale-free behavior of the inverse function is indirectly provided by Moonlight et al. [18] for genera per family.

Past work has dealt with more than twenty-five minimally monophyletic groups [1], and, significantly, the average number of descendant species per ancestor is about four in recently evolved groups not yet subject to much extinction. Also, the number of traits that are newly evolved (generated and fixed in speciation) is about four. Apparently, a Rule of Four [19] involving four as an optimizing factor is present in many fields [20,21,22]. In evolutionary taxonomy, this implies a quadratic fractal process [23,24,25,26] where four is an optimal number that applies across scales. At first, this appeared to be due to crowding [22,27], but it is more probably due to Bayesian exclusion of two-sigma of uncertainty (as discussed below).

The optimal number of species per genus seems to be about 4. Analysis, given time available for revisionary analysis, has stopped (by publishing) at (1) one large genus of 23 species (Leptodontium) split into 10 MMGs [1], (2) three MMGs obtained by resurrection from a larger molecular genus that combined them [28], and (3) 15 other genera of smaller size that proved to be MMGs simply awaiting dissection into progenitor and descendant species [1]. Very large genera like the moss Syntrichia [29] and Didymodon [30] have not been well studied, but are probably either large undistinguished sets of MMGs or obey a different emergent fractal process [31,32], perhaps involving powers of two or four. The large moss genus Chionoloma, however, is a composite of several minimally monophyletic genera [28]. Given the range and number of minimally monophyletic taxa studied, it is thought sufficient to make broadly predictive hypotheses applicable to the large number of already small classical genera, pending work on larger genera.

To be avoided is the multiplication of traits through supervenience, which is the recognition of traits that are not causative. For instance, what are apparently three traits, increase in length, width and depth, may be one trait, an increase in volume. In addition, there may be important traits in addition to the morphological, including those of physiology, ecology, geography, chromosome counts, chemistry, and genomics. However, each of these topics is at a different scale, and, as with the scales of the taxonomic hierarchy, one might expect self-similar numbers of traits at each scale, something definitely worth investigating.

Since graphs of genera per family also exhibit about the same inverse function [33,34], a fractal process may be involved with one generative element (the progenitor) and about four products of speciation (the immediate descendants). The published inverse functions generally follow Equation (1), where the power of the denominator is the fractal dimension of ln 5/ln 4. The fractal dimension describes the common speciation outcome that one ancestor generates about four descendant species; in that all five species share the same novon of the ancestor, generation of five from four is a fractal dimension of ln 5/ln 4, or 1.161.

There are major differences between this method and that of modern cladistics, but the most important is familiarity with the morphological traits of species, which requires taxonomic expertise developed through revisionary work. Happily, taxonomy [35] builds on 250 years of the work of others, and in most groups of organisms the species are at least preliminarily sorted by overall similarity. This sorting eliminates a great deal of the homoplasy (convergent traits) in morphological data sets by making the data sets hierarchically smaller. So small, in fact, that most newly evolved genera are already between 1 and 5 species in size [15]. For revision, the standard dichotomous key, which sorts species names into couplets, is sufficient and helps distinguish smaller sets of species within larger genera. One must be aware, however, that distinctive species are often “taken out” at the start of dichotomous trees to simplify quick identification, and are not necessarily associated with related, less prepossessing species. In addition, fully dichotomous sorting is sufficiently ingrained in taxonomic methodology that it biases the construction of evolutionary trees such that complete dichotomy is viewed as “fully resolved”, as a stopping rule, ignoring the potential of multiple descendant species from one progenitor.

The idea is to recognize minimally monophyletic groups that differ internally by only a few traits between ancestral and descendant species. Identification keys are generally valuable for digesting lengthy descriptions of organisms into the few traits that best distinguish any one species from similar species. A phyletic key requires the recognition of evolutionary relationships and often relies on technical characters. It differs from a standard key in diagraming evolutionary relationships by small or at least parsimonious increments in trait changes. Generalist species are placed at the beginning of the phyletic key and the most specialized at the end, the opposite of standard dichotomous keys, which differ in commonly identify the least specialized species as that which is left after easily distinguished species are identified. The small size of the minimally monophyletic group (MMG) helps instigate the inferential method of using a second-order Markov chain. For this, two items of data are required to establish placement in a phyletic key. The genus ancestral species is that which is (1) most similar to some outgroup species (which may or may not be its own ancestor), and also (2) is most generalist to the ingroup of more specialized species. This is a straightforward simplex-convex situation. Given that, when developing a phyletic key, one ancestral species does commonly generate more than one descendant species, a true phyletic key is necessarily polychotomous.

The criteria for polarization of trait changes are many. Polarizations provide considerable support for a decision in high-resolution phylogenetics because they build sequentially a caulogram by two comparisons (the one species with most similarity to an outgoup, and also most generalized towards ingroup) for determining the identify of a progenitor of multiple descendants. Every MMG has distinctive polarizations of traits involved in its structure. Some polarizations used in revision of the MMGs of this paper include: (1) reduction in size or organ expression; (2) no reversals; (3) parsimony of trait changes; (4) specialization or elaboration; (5) asexual reproduction; (6) precinctiveness or adaptation to small niches or patches; (7) recency of geologic area; (8) isolation of range or geography; (9) chromosome number change; (10) apophyly-paraphyly in molecular analyses; (11) major distance on a molecular cladogram; and (12) rarity of traits. The stopping rule must be the point at which additional traits add little to the resolution of a caulogram, particularly redundant traits that come from different scales when analysis requires total evidence.

A cladogram is simply a standard dichotomous key to species rendered in large part phyletic by cluster analysis of trait changes. The dichotomous nature of an identification key is ingrained in the practice of taxonomy as immensely useful, but is inappropriate for evolutionary resolution of multiple descendant species generated by a single progenitor, even if the progenitor mutates in DNA without a significant change in taxonomically important expressed traits. Expressed traits are the working substance of evolutionary change.

A polychotomous phyletic key is essential to estimating structural monophyly at the microgenus level because this is the first step in how an MMG is distinguished. A microgenus is defined as one progenitor plus its immediate descendant species, each species of the genus sharing the newest traits of the progenitor. The MMG governs how survival-based traits are preserved and shared. A species in ancestor-descendant analysis may have any definition that works within the definition of a microgenus.

The MMGs are concatenated into a larger multi-genus evolutionary tree with much the same criteria. The progenitors of the microgenera (individual phyletic keys) are connected with other progenitors of microgenera by the smallest number of trait changes. Inferentially, this is simple parsimony extended from intra-genus to inter-genus. The caulogram may be a branching dendrogram with names of extant (or rarely inferred extinct) species assigned to the nodes, or it can take the form of a spreadsheet with data embedded in arranged cells. A study of species in the West Indies [22] shows clear evidence of the generation of four descendant species from one ancestral species (Figure 3).

Morphological cladistics has had a bum rap because of the misuse of nonparametric bootstrapping [36] as a method of evaluating consistency and robustness of the data. Cladistic studies rely on large data sets, which in cladistic systematics are evaluated with various optimization methods to determine common ancestry. Parsimony is a basic method, but the use of molecular data has created a cottage industry in complex statistical methods [37], such as coalescent Bayesian Markov chain Monte Carlo. Non-parametric bootstrapping with morphological data is essentially resampling with replacement, a common statistical practice. Because the data sets for morphology are generated from descriptions of species, however, deletion of data for resampling eliminates major features of the species. A species description is a distillation of multiple samples, and the correct application of bootstrapping would be resampling among descriptions of specimens, not species. Resampling descriptions of multiple specimens would strongly support the consistency and robustness of morphological descriptions. To obviate this long-ingrained bias against using morphology in evolutionary systematics, a new method called Shannon-Turing sequential Bayesian analysis [1] is used, which generates posterior Bayesian probabilities for a caulogram structure as high as those for cladograms from molecular analysis.

The method is to assign one Shannon informational bit to each new trait of a newly evolved species. Because bits are logarithmic (base two), they are additive. An odds table (Table 1) provides the Bayesian posterior probabilities [38]. The bits can be added along any short or long stretch of a caulogram, providing support for any number of concatenated species in a lineage. Positioning species on a dendrogram such that there is a minimum or absence of reversals avoids negative bit values, a kind of parsimony. Given the small size of minimally monophyletic groups, the mathematics is straightforward. Conjugate priors can be calculated such that probabilistic support for and against a particular caulogram structure can add to probability 1.00, but this is generally not necessary. Secondary descent (one descendant being ancestor to one other descendant) is treated statistically separately as a separate process, as an inchoate or remnant minimally monophyletic group, which avoids normalization in calculating conjugate priors.

Because morpho-species are already arranged taxonomically in hierarchically nested groups, large data sets are unnecessary for Shannon-Turing analysis (Figure 4). This built-in stopping rule consists of the critical traits involved in speciation being clearly set out when a small, minimally monophyletic group is evaluated for ancestor-descendant relationships. Generally, only three to five traits are involved among those possible character states identified in prior taxonomic studies. The model of evolution in MMGs may be better understood as a simplex-convex model, an analytic technique not yet formally pursued.

Below is a polychotomous phyletic key [28] for the pottiaceous moss genus Pseudosymblepharis. The indentations represent one or more speciation events from one ancestral species. Thus, P. angustata has three descendant species. A new trait is a bit; the odds table (Table 1) gives sequential Bayesian posterior probability [38] as the bits are added along the lineage. Information in this which is essentially a caulogram in sentences, includes a number of novel traits in the speciation event, species name, number of bits in the speciation event, plus that of the previous speciation event used to calculate sequential Bayesian posterior probability (SBPP).

1. Three traits: Neotrichostomum crispulum, 3 bits, SBPP 0.89.

               2. Seven traits: Pseudosymblepharis subduriuscula, 3 + 7 bits, 0.999 SBPP.

                              3. Two traits: P. burneense, 7 + 2 bits, 0.998 SBPP.

               2. Four traits, P. angustata, 7 + 7 bits, 0.999 SBPP.

                              4. Three traits P.dubia, 4 + 3 bits, 0.992 SBPP.

                              4. Two traits, P. orthodonta, 4 + 2 bits, 0.985 SBPP.

                              4. Two traits, P. richardsii, 4 + 2 bits, 0.985.

               2. Five traits, P. schlimii, 7 + 5 bits, 0.9998 SBPP.

                              5. Two traits, P. circinata, 7 + 2 bits, 0.998 SBPP.

                              5. Three traits, 7 + 3 bits., 0.999 SBPP.

The average number of immediate descendants in a minimally monophyletic groups is about four. According to the odds chart (Table 1), four Shannon bits is about 0.941 Bayesian posterior probability. This excludes about two standard deviations (two sigma) of uncertainty. The same holds true for the average number of traits in a species. This value matches the inflexion point of a bell-shaped (Gaussian) curve and is not arbitrary. The process of excluding about 95% of uncertainty is apparently almost universal [39] in evolutionary processes, particularly those reflecting the Pareto ratio, which is involved in the fractal dimension of evolution [22]. The broader implications of a widespread two-sigma exclusionary process will be discussed in a future paper. Selection-mediated exclusion of two-sigma of uncertainty may generate and sustain biodiversity. When established for a taxon, it would provide a strong stopping rule for evolutionary taxonomy. That which is measured as uncertainty is the failure of individuation (survival as a taxon) due to too few traits per species (or species per genus) or lumbering adaptability with too many new traits.

The traits that are newly evolved in speciation are associated with and identify adaptive trends in evolution [40]. This is as opposed to morphological traits that are only mapped on molecular trees generated with common ancestry analysis. Although parsimony is clearly involved in evaluating an MMG, it is done in the context of descent with modification, not common ancestry, and as an optimizing criterion: the most parsimonious MMG is a direct model of expressed traits evolution. Evolutionary trends, that is, traits associated with particular niches or environmental changes, can be confidently ascribed to microgenera. For example, in the moss family Streptotrichaceae, the microgenus Crassileptodontium is clearly adapted to volcanic substrates, while Stephanoleptodontium is strongly associated with both soil and humus at high elevations in the tropics, while the most primitive (first) of extant progenitors, Streptotrichum and its sibling genus, Trachyodontium, are both arboreal [1]. The hypothesis that early genera of the family Streptotrichaceae survived extinction events while living on tree branches is then directly supported, rather than mediated by molecular tracking. Ideally, a molecular study of apophyly and paraphyly should be done to confirm or clarify the morphological results, but the data are as yet unavailable. The strong statistical support for the Streptotrichaceae caulogram [1] implies that predictions regarding evolutionary trends are justified.

That which descends is the set of proven traits of the immediate ancestron, while that which is modified are older traits less important for immediate survival. Conclusions are that one large group of minimally monophyletic genera (Streptotrichaceae of 10 extant genera plus 3 inferred as extinct) is very old in origin, reaching back to the late Cretaceous, while a similarly large group (Pleuroweisieae of 11 extant genera) apparently reach back only to the Paleocene-Eocene era. The range of geological age is correlated with the stability of habitat, and otherwise with exposure to catastrophic environmental change. Estimated geological time is given in Figure 3 for the Neotrichostomum group in the West Indies [22], and in Figure 5 for the family Streptotrichaceae.

The stopping rule for speciation pace in geologic time relies on three criteria: (1) fossil anchoring species of similar morphology, (2) the confinement of one genus of five species to a recent (46 my) land mass, the West Indies, and (3) similar distributions of similar-sized taxonomic groups of similar morphology across the same geologic time scale. The stopping rule is pre-determined by the previous success in using anchoring fossils and an assumed molecular clock, while a new criterion is the dating of available geographic substrate that allows a morphological clock matching to a large extent a molecular clock [28].

A paper [2] has been recently published expressing modes and tempo of evolution in terms of classical mechanics. The number of trait changes is equivalent to evolutionary distance, the expressed character set of any one species is mass, momentum is distance times mass, and force is acceleration times mass. By equating force with evolutionary resilience to environmental change, each microgenus may be assigned a measure of success in sustainability over geologic time scales. In this study, the innovation is that by adding all new trait changes along a caulon (a single line of stem-taxon species from base to species), one can place a comparative measure on acceleration. If the mass is considered the number of species in the caulon (adding all progenitors in a line), then force, as acceleration times mass, is easily calculated. The stopping rule is the ability to make comparisons between species, genera, and higher taxa in regard to their differential evolutionary resilience with similar results in two different bryophyte lineages, the family Streptotrichaceae and Pottiaceae trube Pleuroweisieae (Figure 6).

The fact that a progenitor shares its most recent traits with all immediate descendant species imbues the microgenus a redundancy of traits. That is, in a microgenus, all species share an identical set of ancestral traits. Such redundancy has evolutionary importance [41] in providing flexibility in a group. The redundancy of traits in microgenera can be expressed as probabilistic information, this in terms of Shannon entropy [5,42]. Redundancy is calculated with Shannon methods [4], where maximum entropy minus Shannon entropy divided by maximum entropy as a normalizing function yields a redundancy measure.

A comparison of these measurements in more than 20 microgenera [5] demonstrated that these values were all very similar across all microgenera. A portion of the spreadsheet is given here in Table 2. The implication is that a natural process, probably natural selection, has fixed the level of trait redundancy to be similar in all MMGs. A lineage is optimized for survival by natural selection and descent with modification and, through redundancy has a self-correcting structure. The stopping rule is that the same large number of sampled species have trait composition that allows predictive hypotheses to be extended to other small genera and perhaps to other large genera that are potentially able to be dissected into MMGs.

High-resolution phylogenetics is a form of numerical taxonomy that melds classical evolutionary taxonomy with molecular systematics and reaffirms the importance of standard taxonomic methods. Polychotomous and multiorder Markov chains are more consonant with evolutionary theory than a cladogram’s dichotomous and first-order Markov chain. Standard practice for taxonomically evaluating variation in a species, short of fully fledged statistical analysis, is to continue sampling and not halt until there is no change; that is, the range of states is established, given reasonable effort and an expectation that most species are not entirely stenomorphic. This is not done in molecular analysis, which no longer has the excuse of DNA sampling being too expensive. The variable that is critical for high-resolution evolutionary analysis is not that which tracks shared ancestry, but that which reveals apomorphy-paraphyly polarizations. Following is a numbered list of the more salient criticisms of cladistic phylogenetics.

Against such practice, a recent analysis of the plastome of the genus Tortula [61] has shown strong purifying selection in most genes, and positive selection in a smaller number, based on distribution of nonsynonymous-to-synonymous substitution ratios. A recent study of the large pottiaceous moss genus Syntrichia [29] used 354 nuclear exons for each of 79 species. The resulting molecular cladogram placed many species into ecologically similar groups. An appraisal of one of the terminal groups of several arboreal species showed no morphologically based structural monophyly, that is, no simple arrangements of closely related species in small groups that might be interpreted as minimally monophyletic. The use of exons is ill-advised due to the probability of massive bias due to convergence.

Given the above caveats and possible problems, a simplified set of steps for the structural monophyly analysis is given below:

The aim of high-resolution phylogenetics using morphology in evaluating structural monophyly is to identify and clarify the processes of evolution that have led to present-day robust, long-term, resilient biological lineages that support shared ecosystems. Biodiversity is generated and stabilized by the minimally monophyletic group in its sharing of the new traits of the ancestral species with all its immediate descendants, and the quadratic restriction of numbers of descendant species, apparently important at all scales, given self-similarity [62] at genus and family levels.

It may be suggested that genera provide the low Shannon entropy of the immediate ancestron’s massive redundancy, supporting survival of descendant species in present-day environments. This is while higher level taxonomic groups of tribes, subfamilies, and families each present a bank of survival-tested character states in their character sets, of which one of each character state is active and expressed somewhere in the lineage for immediate adaptive action on behalf of the lineage during environmental change, while the remaining potential character states are available for more longer selection and fixation processes. Thus, genera are well-equipped to survive locally in space and time, while the lineage of several to many minimally monophyletic groups holds the keys to long-term survival of the entire assemblage, including an unexpectedly fast rate of evolution in response to rapid environmental change.

The stopping rules for structural monophyly analysis require thorough analysis of variation in both morphology and molecular sequences, identification of the basic minimally monophyletic group using second-order Markov reasoning, a matching of morphological and molecular information on ancestor-descendant relationships (apophyly-paraphyly), and high statistical support for both morphological and molecular hypotheses of descent with modification, through matching analyses of total evidence.

If the original pre-determined stopping rule for evolutionary analysis is to generate a well-supported evolutionary tree using total evidence, then cladistics stops too soon. The original stopping rule is simply abandoned in favor of a molecular-only ultrametric or phyletic cladogram where polychotomies are deprecated as “unresolved”. The existence of an alternative method that more accurately reflects the aims of a reasonable stopping rule should encourage an abandonment of molecular-only cladistic cluster analysis and the mapping of morphological traits to molecular cladograms.

During the preparation of this manuscript, the author used Gemini/Google for literature searches. After using this tool/service, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article.

The author is grateful to the Missouri Botanical Garden for its continued support of novel research in plant systematics.

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All relevant data are included in the article.

This research received no external funding.

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.