A laid-back trip through the Hennigian Forests

Introduction

Here it would seem more appropriate to re-write the characters …. in a tree-form representing the relationships exactly

Williams & Ebach (2006: 414)

... synapomorphies might evolve more than once. Suppose, indeed, that all synapomorphies really have multiple origins. Would our understanding of relationships then be any different than it now is?

Nelson (2011: 140)

Matrix-free Cladistics (Platnick, 1993; Williams & Ebach, 2006; Williams & Ebach, 2008; Zaraguëta-Bagils et al., 2012) is a way to represent the binary, morphological and molecular data in tree form, therefore dispensing with the traditional (ordinary) matrix. Demonstration of this idea’s efficiency is a major goal of the study.

Recently, the matrix-free Cladistic approach has been developed by Zaraguëta-Bagils et al. (2012). Our framework, however, is different from their proposal, which we discussed below. In general, we are viewing the method developed by Zaraguëta-Bagils et al. (2012) as their original conceptualization of the three-taxon statement analysis (3TA) (Nelson & Platnick, 1991) which is essentially based on the principle of the maximum compatibility (see Wilkinson, 1994a for the initial discussion).

One may note that our vision of the topic replaces the standard “processes-based” subject with a purely geometric view of the problem. Another may see that our viewpoint is closely related to the old discussions around “Transformed cladistics” (Platnick, 1979; Platnick, 1985) or “pattern-cladistics”, or “Cladistics” (with the capital “C”, see for example Hull, 1988). Numerous works have been published on topic in the past (e.g., Betty, 1982; Brady, 1982; De Queiroz & Donoghue, 1990; Nelson, 1989; Nelson & Platnick, 1981; Patterson, 1980; Patterson, 1982; Platnick, 1979; Platnick, 1982; Platnick, 1985 see also Brady, 1985; Brower, 2000; Eldredge & Cracraft , 1980; Ebach & Williams, 2013; Nelson, 1972; for the detailed reviews of the topic see Hull, 1988; Farris, 2014, Scott-Ram, 1990 and especially Williams & Ebach, 2008). We would like the reader to make his own conclusions regarding the actual summary of these mostly aged but still inspiring debates, which in our minds, were terminated artificially. Of course, it would be impossible to reproduce anything but the shadow of that story in this Introduction.

However, today it is still critical to repeat that dealing with the ‘trees’ (or with the statements of relationships) can be viewed separately from any considerations of the process (e.g., Kitching et al., 1998: p. 1; Patterson, 1980: p. 239). Also, the statements of the relationships may be easily treated just as invariants of the processes (either real or hypothetical). For example, when considering the statement A(BC) it is easy to conclude that it does not really matter what the actual cause or process that linked taxa B and C altogether was.

We agree that the consistent implementation of the basic idea of Hennig—grouping solely on synapomorphies returns us to purely comparative thinking (e.g., Nelson, 2004; Williams & Ebach, 2008), even in the case of molecular data. From that, the treatment of the standard optimization procedures, performed either under the maximum parsimony or other criteria, as something that is related to the reality with its processes, appears to be very naive. In our mind, switching from the matrix-based thinking to the matrix-free Cladistic approach clearly reveals that the optimizations of the character-state changes are related not to the reality per se, but to the form of the data representation.

The aspects of the Cladistics we approved are clearly summarized in Patterson (1980):

“Hennig’s 1966 book, as the title Phylogenetic Systematics suggests, was based in evolutionary theory…But as the theory of cladistics has developed, it has been realized that more and more of the evolutionary framework is inessential, and may be dropped. The chief symptom of this change is the significance attached to nodes in cladograms. In Hennig’s book, as in all early work in cladistics, the nodes are taken to represent ancestral species. This assumption has been found to be unnecessary, even misleading, and may be dropped. Platnick (1979) refers to the new theory as “transformed cladistics” and the transformation is away from dependence on evolutionary theory. Indeed, Gareth Nelson, who is chiefly responsible for the transformation, put it like this in a letter to me this summer: “In a way, I think we are merely rediscovering pre-evolutionary systematics, or if not rediscovering it, fleshing it out” ”

(Patterson, 1980: p. 239; italics ours).

As Nelson himself defined later, Cladistics is “ordering homologies into a parsimonious hierarchy, which specifies both taxa and their characters (apomorphies)” (Nelson, 1989: 279), and therefore it is independent from the principle of common descent (Nelson, 1989).

Among others, De Queiroz & Donoghue (1990) have criticized this vision of the topic. In our mind, their most important contra-arguments are:

1. Nelson’s vision of cladistics is strongly related to the “model of nested hierarchy”, even if they are supposedly independent of the principle of common descent (De Queiroz & Donoghue, 1990: p. 62);

2. the conventional phylogenetic systematics has greater explanatory power than that which underlies what Nelson calls “cladistics” (De Queiroz & Donoghue, 1990: p. 61, see also Farris et al., 1995 and Farris, 2013).

The later criticism of the transformed cladistics has been developed by James Farris and his school (see Farris et al., 1995 and Farris, 1997; Farris, 2011; Farris, 2012; Farris, 2014 for the discussions and the summaries of the arguments), essentially in relation to the possible issues of the three-taxon statement analysis (Nelson & Platnick, 1991), that we, contrary to Brower (2000), are viewing as the first analytical implementation of the transformed cladistics approach (e.g., Kitching et al., 1998). Below we show that the major empirical claims against 3TA, established by Farris and co-authors (see Siebert & Williams, 1998, Mavrodiev, 2015b and Mavrodiev, 2016 for the reviews), are not relevant to our recent framework.

In our mind, the whole focus of the “explanatory power” (e.g., De Queiroz & Donoghue, 1990; Farris et al., 1995) is essentially misleading due to its strong relation to the particular philosophy of science that different authors are following or implying. For example, this focus is not quite relevant within the framework of critical philosophy (Mavrodiev, 2016, see also Van de Vijver et al., 2005 and Kolen & Van de Vijver, 2007 among others), which has likely never been mentioned in the discussed context until recently.

B. Initial propositions

A binary character is a tree with one informative node (Platnick et al., 1996; Williams, 1994; Williams & Ebach, 2006; Williams & Ebach, 2008). For example, if state 1 is apomorphic, then the character ABCDE/00011 is a rooted tree ABC(DE), where (DE) is a clade (monophyletic group) based on the apomorphic character-state. If all of the clades of the tree are based on apomorphic character-states, we call this tree a “Hennigian” tree. If we accept that the trees such as ABC(DE) or A(BC(DE)) are Hennigian, than the trees A((BC)(DE)) or ((ABC)(DE)) are not. The non-Hennigian trees contain groups based on plesiomorphic characters-states—for example, tree ((ABC)(DE)) is non-Hennigian because group (ABC) based on the plesiomorphic characters-state zero. Hennigian trees may also be easily seen as simple hierarchies of two character-states.

Let A be defined as the outgroup. In this case the relationship ABC(DE) may be re-written as A(BC(DE)) (or as a (A(BC(DE)))), even if, strictly speaking, we do not have any formal evidence for the groups (BCDE) (or (ABCDE))v (see Rieppel, 1991: p. 95 and Williams & Ebach, 2009 for the related discussions). For example, the minimal relationship A(BC)/011 may be re-written as (A(BC)), even if there is no evidence for the group (ABC).

Consider the tree A(BC(DE)) for one more time. If the value of taxon B is missing, two solutions appear to be possible.

First, character ABCDE can be re-written as two trees, assuming that the missing value may be either zero or one: ABC(DE) = A(BC(DE)) or A(C(BDE)). Another possibility implies the exclusion of the taxon B from the tree. This reduces the character ABCDE to the tree AC(DE).

Characters like ABCDE/00001, 000?1, ??001, 00000 etc. collapse to polytomies such as (ABCDE), (ABCE) etc.

The basic idea of Matrix Representation with Parsimony (Baum, 1992; Ragan, 1992) states that the tree (cladogram) can be represented as a binary matrix. Here we propose something opposite—we accept that the binary character may be re-written as a tree (Nelson & Ladiges, 1992, 492–493; Williams, 1994; Siebert & Williams, 1998: 342; Williams & Ebach, 2006: 414; see also Platnick et al., 1996). Therefore, the binary matrix can be viewed as a set (forest, array etc.) of branching diagrams (cladograms) that may be called a “Hennigian forest” if one of the character-states (for example, the character-state zero) a priori is defined as plesiomorphic.

It is easy to assume that the Hennigian forest might be analyzed at least by some consensus methods, typically treated as a ways of estimating the supertrees (Gordon, 1986, see also Wilkinson, Cotton & Thorley, 2004 and Bininda-Emonds, 2014 for the reviews) for example, by the Average Consensus method (Lapointe & Cucumel, 1997; Lapointe & Levasseur, 2004, see also Bininda-Emonds, 2014). Using this method as an example, below we are attempting to display the workings of this assumption.

As summarized by Lapointe & Levasseur (2004: 87), the Average Consensus procedure is a method that takes as input a profile of weighted trees (i.e., trees with the branch lengths Lapointe & Cucumel, 1997; Creevey, 2004; Lapointe & Levasseur, 2004) and returns a consensus tree that is, in some sense, “closest” to the entire profile. Originally this method was designed to combine the clock-like or “ultrametric” rooted trees (or the trees with the total branch length from the root up to any tip equal (Felsenstein, 2004: p. 161)), but later have been extended to allow for the combination of all types of the weighted trees, the clock-like or not (reviewed in Lapointe & Levasseur, 2004: 88 and Creevey, 2004: p. 18).

This method allows us to operate with the Hennigian trees directly, completely excluding the binary data matrix from the analysis. The ordinary (phenetic) matrix remains necessary only as a “source” of the forest of the Hennigian trees, not as a primary subject of the analysis. Within this framework, the criteria of the best trees, such as the minimal numberof character-state changes, as well as the standard optimization procedures, if optimization is defined as a way to reconstruct the sequences of the character-state changes on a tree, all appeared to be unrequired.

The matrix-free Cladistic approach, as designed and implemented by Zaraguëta-Bagils et al. (2012) and others (reviewed in Zaraguëta-Bagils et al., 2012), also treats the characters as hierarchies or as rooted trees, but clearly differs from our proposal in several aspects. According to the logic and implementation of Zaraguëta-Bagils et al. (2012), the rooted character-state trees must first be reduced to a set of three-item statements. The software LisBeth (Zaraguëta-Bagils et al., 2012) performs the modified three-taxon statement analysis, which is based on the maximum congruence of the statements and finally calculates the intersection that the tree is built from, and only from, the three-taxon statements that are common to all the optimal trees and characters (Zaraguëta-Bagils et al., 2012).

Our approach also differs in principle from the numerous purely phenetic alignment-free techniques of the construction of the phylogenetic trees (reviewed and summarized in Warnow, 2014 and Bogusz & Whelan, 2016) because these methods do not even refer to the concepts of the “relation” and “homology”, but operate by computing similarities (the pairwise distances) between the raw (or non-aligned) sequences (e.g., Warnow, 2014).

Finally, one may ask why the data cannot be imputed directly as trees (Zaraguëta-Bagils et al., 2012) or, in other words, by just completely skipping the data matrix? In principle, it is possible. However so far this solution seemed for us to be impractical in many cases.

Materials and Methods

Despite the plethora of publications within the field of contemporary Bioinformatics, there is no software available to rewrite the binary matrix as a Hennigian forest. Given this, we wrote the simple ruby-based script FORESTER version 1.0 (named “FORESTER” below) that helps represent the binary data as Hennigian trees for future manipulations (Figs. 1–9).

FORESTER (deposited on https://github.com/dellch/forester) processes each input file by storing the text of each line of the file as an element of an array. It finds the beginning and ending location (index) of the characters in the matrix, and loops from the starting index to the ending index, writing a line of the new file(s) at the end of each loop.

For the binary Matrix, the usage of the script is: ruby trees.rb –inputfilename

The pre-defined out-group taxon should be placed last in the matrix before running of the FORESTER.

Three output tree-files are available as a result of the run: the first file contains polytomies such as (ABCDE)(the output tree-file named as a “With poly…” ), while the second and the third outputs appear without such polytomies, but trees may be rooted relative to a priori-defined outgroup taxon (e.g., (A(BC(DE)))) (the output array appears as named as “No poly…” tree-file) or to the basal polytomy (e.g., (ABC(DE))) (the output tree-file named as an “Additional…” tree-file).

All tree files and the input binaries should be written in the “relaxed” non-interleaved Newick (PHYLIP) format (reviewed in Felsenstein, 1989, see also Maddison & Maddison, 2011 and Swofford, 2002).

The minimal trees (the three-taxon statements (3TS)) are not the subjects of our recent considerations, but FORESTER contains options for speedy rewriting of the 3TS matrices as the arrays of the minimal trees. The usage should be: ruby seedlings.rb –inputfilename with only one output Newick file saved as a result of the run.

Seven binary matrices (Figs. 1–4, 5, 9) are taken from the literature (Nelson, 1996; Kluge, 1994; Farris, 1997; Farris & Kluge, 1998; Kluge & Farris, 1999).

Preparing the DNA supermatrix of the multiple loci of the Great Apes (Hominidae, Primates, Mammalia) (Fig. 6) we first used MUSCLE (Edgar, 2004) to align separate 98 gene regions listed in the modified summary of the Lehtonen et al. (2011). The resulting separate DNA alignments were concatenated in the SeaView 4.0 (Gouy, Guindon & Gascuel, 2010) after removing purely aligned regions using the Gblocks 0.91 (Talavera & Castresana, 2007) as implemented in a Seaview (Gouy, Guindon & Gascuel, 2010). The Gblocks less stringent selection strategy (Talavera & Castresana , 2007) was decided to be the best choice. Eventually, the only characters with non-ambiguous value of the outgroup (Papio) have been saved within concatenated 42,222 bp alignment.

Matrix of 166 muscular characters of Primates (Mammalia) and Rattus (Mammalia, Rodentia, Muridae) has been modified from the image of Diogo & Wood (2012: p. 121, Table 3.1). We treated all multistate characters from Table 3.1 (Diogo & Wood (2012: p. 121) as ordered (reviewed in Kitching et al., 1998) (Fig. 7) and recoded all of them in additive binary form (reviewed in Kitching et al., 1998) using TAXODIUM ver. 1.2 (Mavrodiev & Madorsky, 2012).

The 15,9074 bp alignment of complete plastomes of palms (Arecales), Dasypogonales, and Typha latifolia (Poales, Typhaceae) (Fig. 8) has been downloaded from the online supplement of Barrett et al. (2016: their Fig. 4 and Fig. S5).

The binary representations of the genomic (DNA) data (Figs. 6, 8 and 9) have been made by script 1001 (Mavrodiev, 2015a). In theory, different binary representations of the DNA alignments (Mavrodiev, 2015a) can be chosen as sources for the Hennigian trees. But in this paper, for future analyses, we decided to select the simplest binary matrices that resulted the “presence–absence” recoding (reviewed in Kitching et al. 1998) of the DNA alignments (Mavrodiev, 2015a), but saving the only characters that corresponded to the value of zero of the outgroup taxon. Script 1001 (Mavrodiev, 2015a) is the easiest way to generate such binary matrix. However, it can be prepared even manually using the text editor.

The Average Consensus tables are calculated following the default settings of Clann version 4.1.5 (http://chriscreevey.github.io/clann/) (Creevey, 2004; Creevey & McInerney, 2005; Creevey & McInerney, 2009) for the avcon command with the branch lengths of the all input trees assigned to unity (Creevey, 2004) and also used as future inputs for PAUP* 4.0b10 (Swofford, 2002) (PAUP* hereinafter). All input-trees are treated as weighted equally.

In most of the analyses the optimality criterion of the best tree is defined in the simplest way—as a distance with non-weighted least squares (Creevey, 2004; Creevey & McInerney, 2005; Creevey & McInerney, 2009; Swofford, 2002). In the case of the analysis of the Hennigian forest of Matrix Z (Farris, 1997) (Fig. 5E–5H) we used the weighted least squares (exponential weights with P′ = 7 (Swofford, 2002)). Different formal criteria, such as “Balanced Minimal evolution” (BME) (Desper & Gascuel, 2002; Swofford, 2002) (Figs. 7B and 7C) can also be used.

All standard Maximum Parsimony (MP) analyses as well as Maximum Parsimony Bootstrap (MP BS) estimation (Fig. 6) have been conducted in PAUP* (Swofford, 2002), sometimes (Figs. 7 and 8) as implemented in CIPRESS (Miller, Pfeiffer & Schwartz, 2010) using 1,000 random addition replicates (saving no more than 100 trees per replicate), and with the TBR branch swapping/MulTrees option in effect; the gaps were treated as “missing entities”.

The routine manipulations with the matrices and the tree-files were performed with Mesquite v. 3.01 (Maddison & Maddison, 2011), PAUP* (Swofford, 2002) and FigTree v. 1.4.2 (Rambaut, 2012).

Results and Discussion

In Cladistics, clades must be based solely on “derived” or apomorphic character-states (e.g., Hennig, 1966; Platnick, 1985; Nelson, 2004; Williams & Ebach, 2008). For example, in the case of the binary matrix, where the character state “zero” is defined as “plesiomorphic” before the analysis, all clades should be based solely on the state “one” (e.g., Platnick, 1985). However, today the 3TA (Nelson & Platnick, 1991) is the only method, which completely avoids grouping on plesiomorphy (e.g., Williams & Ebach, 2008). Therefore, analyzing all of the selected matrices (Figs. 1–9) we have followed the general logic of 3TA. Specifically, (1) we tried to explicate all possible Hennigian trees a priori to the analyses in order (2) to find the best-fitting trees as the next step.

Five out of the seven binary matrices-examples (Figs. 1, 3, 4, 5) we analyzed have been established as a main sources of the major empirical claim against 3TA (and in our mind, against the whole idea of grouping solely on synapomorphies)—the principle inability of this method to operate with the putative reversals (Kluge, 1994; Farris, 1997; Farris & Kluge, 1998 reviewed in Siebert & Williams, 1998, Mavrodiev, 2015b and Mavrodiev, 2016). Using these critical examples we are demonstrated (Figs. 1, 3, 4 and 5) that this claim is not relevant to our recent proposal. Also using the original critical example of Kluge & Farris (1999) we also showed that our method could successfully handle the non-clock molecular data (Fig. 9).

We are not viewing our recent approach as an alternative to 3TA. However, there are two major differences between our recent proposal and the 3TA as originally designed and implemented by Nelson & Platnick (1991). (1) The 3TA deals with the three-taxon statements (3TS) or minimal relationships (Williams & Siebert, 2000; Williams & Ebach, 2006; Williams & Ebach, 2008). The set of 3TSs must be established as a binary matrix and used as an input for the parsimony program (Nelson & Platnick, 1991; see also Williams & Siebert, 2000, Williams & Ebach, 2006. Williams & Ebach, 2008 and Mavrodiev & Madorsky, 2012).

In this paper, we operate directly with maximal relationships (Nelson & Platnick, 1981; summarized in Williams & Ebach, 2006 and Williams & Ebach, 2008) written just as trees, not as binary matrices, while also using the Average Consensus method instead of the MP analysis.

Dealing with relations, not with actual “data”, clearly distinguishes our approach from all optimization-based methods, both the standard alignment-based methods and the “direct” alignment-free approaches (e. g., Wheeler, 1996; Wheeler, 2001).

In summary:

(1) Script FORESTER helps to rewrite every character of the binary matrix as a tree in a Cladistics way basing all of the groups solely on a priori defined apomorphic character-state “1”.

(2) Next, the forest of these “maximal” Hennigian trees can be used to calculate their average consensus (Lapointe & Cucumel, 1997; Lapointe & Levasseur, 2004) (Figs. 1–9).

A possible issue of the Hennigian approach to the data is the inability to operate with the putative reversals (reviewed in Farris & Kluge, 1998, Siebert & Williams, 1998 and Mavrodiev (2015b), unless, however, the ‘reversals’ are scored as separate apomorphic character-states before the analysis (Mavrodiev, 2015b).

Paradoxically, the Average Consensus method, if applied to the Hennigian forests, helps to identify the ‘reversal’-based clades without the separate scoring of the putative reversals, or, in other words, despite the absence of evidence from the primary data (Figs. 1, 3 and 4). A similar effect had been described by Nelson & Platnick (1991) for 3TA and discussed in more detail by Siebert & Williams (1998) as well as by Mavrodiev (2016) who named this paradox as a “Synthetic Theorem of Nelson and Platnick”.

For example, consider the modified Matrix 17 from Nelson & Platnick (1991) (Fig. 1). According to the MP optimization, group (EF) is based on the plesiomorphic character-state zero and therefore may be treated by someone as a putative ‘reversal’-based clade (e.g., Farris, 1997). However as explained above, grouping based on the plesiomorphic character state is prohibited within the Hennigian framework (e.g., Platnick, 1985; Nelson, 2004; Williams & Ebach, 2008). Therefore the standard MP solution for the group (EF) is clearly non-Hennigian or phenetic (e.g., Williams & Ebach, 2008; Mavrodiev, 2015b).

None of the trees from the array c. (Fig. 1) contains any plesiomorphic-based groups, but clade (EF) is still defined in the average consensus tree (Fig. 1) throughout the analysis of forest c. (Fig. 1) or, in other words, despite the lack of evidence from the primary data (see also Nelson & Platnick, 1991: 363 for the similar discussion). The same situation is detected in the cases of Table 1 from Kluge (1994) (Fig. 3), and Table 5 from Kluge & Farris (1999) (Fig. 4).

The Average Consensus technique, if applying to the Hennigian forests, may also help to avoid more potentially negative effects of putative reversals. For example, regarding his Table 3, Kluge (1994: 408–410) mentioned that taxa G and F are highly supported sisters with taxon G exhibiting reversals only in characters one and two. As discovered by Kluge (1994), the 3TA (Nelson & Platnick, 1991) removed taxon G from F, despite the strong evidence of their relationship. A similar situation appears in the case of Matrix Z, designed by Farris (1997) simply by duplication of the Table 3 from Kluge (1994) (Farris, 1997; Siebert & Williams, 1998).

However, the tree-shapes of the average consensuses of the Hennigian forests of the Table 3 from Kluge (1994) (Fig. 5A–5D) and Matrix Z from Farris (1997: 136) (Fig. 5E–5H) appeared to be identical to the topologies, which were the result of the MP analyses of Table 3 and Matrix Z (Kluge, 1994; Farris, 1997). 3TA, with fractional weighting procedure (Nelson & Ladiges, 1992; Williams & Ebach, 2008) can also compensate the negative effect of putative reversals in the same situations (Siebert & Williams, 1998).

Like the 3TA (Nelson & Platnick, 1991), the Average Consensus analysis of the forests of maximal relationships can successfully recognize groups for which the standard optimization criteria of the MP analysis produce no unequivocal synapomorphies (Nelson, 1996; Williams & Ebach, 2005) (Fig. 2). For example, group (BCD) is successfully recognizable after the Average Consensus analysis of the three trees, each representing the conflicting binary characters from Matrix 1 from Nelson (1996) (see also Williams& Ebach, 2005; Williams & Ebach, 2006) (Fig. 2).

Williams & Ebach (2006: 414) offered a very similar solution for the clade (BCD). These authors mentioned that it would seem more appropriate to re-write the characters of Matrix 1 from Nelson (1996) in a tree-form representing the relationships exactly, such that the three “characters” are AD(BC), AC(BD) and AB(CD), which, when “combined” (Williams & Ebach, 2006: 414, italics are ours), unambiguously provide evidence for the solution A(BCD) (Williams & Ebach, 2006: 414). In contrast to this intuitively clear solution, as well as to the results of the Average Consensus analysis (Fig. 2), neither of strict, majority rule, Adams’, or combinable component consensuses (see Kitching et al., 1998 and Swofford, 2002 for the reviews and implementation of the methods) of the three trees AD(BC), AC(BD), and AB(CD) (Williams & Ebach, 2006: 414) are able to recognize the group (BCD).

Williams & Ebach (2006: 414) are also noted that the re-writing of the binary characters effectively converts an ordinary (phenetic) matrix into a “Cladistic matrix”. In our mind, a Cladistic matrix is similar, but still not of the same entity as a simplest forest of the Hennigian trees (see Siebert & Williams, 1998: 242 for the initial elementary examples of the Cladistic matrices called by these authors as a “tabular formulations of the 3TA” of the conventional data; see also Williams & Ebach, 2006: 412, Nelson & Ladiges, 1992: 492–493 and Williams & Ebach, 2008).

In the first approach, the simplest options of the search (the least-square criteria of the fit plus un-weighted algorithms) should be sufficient to estimate the reasonable average consensuses. However, the Average Consensus approach allows various schemes of weighing as well as different optimization criteria (Creevey, 2004; Lapointe & Levasseur, 2004). For example, the use of the exponential weights of the least squares as implemented in PAUP* (Swofford, 2002) or BME-algorithm (Desper & Gascuel, 2002; see (Swofford, 2002 for recent implementation) may increase the efficiency of the search (Figs. 5 and 7).

The proposed approach may also handle the clock-like (e.g., Fig. 6) or non clock-like matrices (e.g., Fig. 9), as well as the hypothetical or the real data, either the DNA sequence data or the morphological characters (Figs. 6–8). There are several possible ways to do this. As a tentative test, for someone it may seem to be necessary to re-recode the conventional multistate matrices as a binary matrices, to establish these matrices as a Hennigian forests, and to estimate the average consensus trees that fit these Hennigian forests (Figs. 6–8). Therefore, even in the cases of the big molecular data (Fig. 8), the taxon-character matrices (molecular alignments) may be necessary only as sources of the arrays of maximal relationships.

We also would like to stress, that for the real data it is also true that at least in some cases the clades may appear within the final tree without the direct evidence from the primary data. For example, the clades (Homo plus Pan) or (Cocos nucifera plus Elaeis guineensis) are still recognizable by the Average Consensus method, even if all of the trees from the corresponding Hennigian arrays, that contain the same clades (Cocos nucifera plus Elaeis guineensis) or (Homo plus Pan), will be excluded from the forest-inputs (Figs. 6C, 7C and 8C). In other words, the clades (Homo plus Pan) or (Cocos nucifera plus Elaeis guineensis) appears within the final average consensus trees (Figs. 6–8) as a kind of relations (e.g., Nelson, 2011), rather than the character-based entities.

Conclusions

1. Binary matrix with a priori defined plesiomorphic character-state may be re-written as a simple set of rooted branching diagrams (as a “Hennigian forest”).

2. The last might be analyzed by the Average Consensus method.

3. This procedure eventually avoids the taxon-character matrix from the analysis of the data. Within the proposed framework, the ordinary (phenetic) matrix is necessary only as a “source” of the Hennigian trees, not as a primary subject of the analysis.

4. Within this approach, the criteria of the best trees based on the character-state changes are not required.

5. Dealing with the relations, not with the actual “data”, clearly distinguishes our approach from all optimization-based techniques, either the standard taxon-character alignment-based methods, or the “direct” alignment-free method.

6. The Average Consensus method, if applied to the Hennigian forests, may helps to identify the clades despite the absence of the direct evidence from the primary data. Because of such ability, the ‘reversal’-based clades can always be found by this approach without the separate scoring of the putative reversals.

Supplemental Information