Tree-metrizable HGT networks

Andrew Francis; Michael Hendriksen

REVIEW 3 major objections 6 minor 10 references

Tree-metrizable HGT networks

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every tree metric can also come from an HGT network with h−1 reticulation arcs.

desk verdict Bold and interesting, but the main theorem's proof has a real gap and the headline result is not yet established. read the letter →

arxiv 1908.08647 v1 pith:OFUPZTUB submitted 2019-08-23 q-bio.PE math.CO

Michael Hendriksen , Andrew Francis This is my paper · ORCID

classification q-bio.PEmath.CO MSC 92D1505C0505C20

keywords treemetricfour-pointconditionHGTnetworkhorizontalgenetransfertree-metrizablecaterpillarreticulationphylogenetic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether distances that satisfy the four-point condition—and therefore fit a unique phylogenetic tree—can also be produced by a phylogenetic network that contains horizontal gene transfer (HGT) events. The authors prove that the answer is yes in a strong form: for every rooted binary tree $T$ of height $h>2$, there is a non-trivial HGT network whose underlying tree is $T$ and that carries exactly the same leaf-to-leaf distances as $T$, while containing $h-1$ reticulation arcs. The construction starts from caterpillar networks, whose display trees are explicitly listed, and extends them by leaf-grafting. The consequence is that satisfying the four-point condition is not evidence against a reticulate evolutionary history: every tree metric is also a metric of some many-reticulation HGT network.

What carries the argument

The load-bearing object is the caterpillar network: a network whose underlying tree is a caterpillar tree (a tree with one cherry, leaves ordered $x_1,\dots,x_n$), with an HGT arc from each leaf $x_i$ ($1\le i\le n-2$) to the last leaf $x_n$. It has exactly $n-1$ display trees $T_1,\dots,T_{n-1}$, which makes the four-point condition manageable. Lemma 3.2 turns tree-metrizability of a four-leaf network into an inequality between probability-weighted sums of internal-arc lengths of the three quartet splits; Lemma 5.3 computes those sums for caterpillar networks; Theorem 5.4 chooses the edge lengths $\ell_j$ so that the required inequalities hold, making the network tree-metrizable on every displayed tree. The Replacement Theorem 4.3, which says that grafting a tree onto a leaf preserves tree-metrizability, then lifts the caterpillar result to arbitrary trees in Corollary 5.5. A second construction, the enhanced caterpillar network, is shown to remain tree-metrizable when grafted onto any tree (Theorem 6.7).

What would settle it

Run the Theorem 5.4 construction on a five-leaf caterpillar network: fix positive values for the internal arc lengths $m_2,m_3$ and for the probabilities $\beta_1,\dots,\beta_4$, then solve the displayed linear equations for $\ell_1,\ell_2,\ell_3$ (including the extra equality connecting $\ell_{i+1}$ and $\ell_{i-1}$). If any choice of positive parameters forces a nonpositive $\ell_j$, the asserted positivity claim fails for those parameters; a computer search over the parameter range would settle whether the proof's claim is universally true.

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Extended reading notes

Core claim

The central claim is that tree-likeness of a distance matrix does not limit the number of reticulation events that could have generated it. For any rooted binary phylogenetic tree $T$ of height $h>2$, Corollary 5.5 constructs a non-trivial HGT network with underlying tree $T$ and exactly $h-1$ non-trivial reticulation arcs that is $T$-metrizable: the network's convex combination of display-tree metrics equals the metric of $T$. The proof goes through caterpillar networks $C$ on $n$ leaves, in which each of the first $n-2$ leaves sends an HGT arc to the last leaf. Theorem 5.4 shows such a network is tree-metrizable on every tree it displays by choosing the edge lengths $\ell_j$ to satisfy a linear system built from Lemma 5.3's formulas for internal-arc sums; Corollary 5.5 then grafts pendant subtrees onto the caterpillar leaves via the Replacement Theorem 4.3 to realize an arbitrary tree $T$ of height $h$.

Load-bearing premise

The construction's load-bearing premise is that, for each display tree $T_i$, the linear system in Theorem 5.4 has a strictly positive solution for the edge lengths $\ell_j$ (the $\gamma_\pm$ equations plus the extra equality); the paper asserts this is a simple exercise in linear algebra without exhibiting the solution or proving positivity. If some parameter choices force a nonpositive $\ell_j$, the caterpillar construction—and with it Corollary 5.5—fails.

Editorial extensions

If this is right

Every rooted binary tree of height $h>2$ admits a non-trivial HGT network with the same metric and $h-1$ reticulation arcs (Corollary 5.5).
Tree-metrizability is preserved when a tree is grafted onto any leaf of a tree-metrizable network, so the class of tree-metrizable networks is closed under adding arbitrary pendant subtrees (Theorem 4.3).
Caterpillar networks are tree-metrizable on every tree they display, so the metric alone cannot single out the underlying tree among a network's display trees (Theorem 5.4).
Networks formed by leaf-grafting an enhanced caterpillar network onto a tree are tree-metrizable, giving a broad positive class for the network-onto-tree grafting question (Theorem 6.7).
Single-reticulation restrictions from earlier work do not extend: tree metrics are compatible with arbitrarily many HGT events.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Corollary 5.5 is right, then the four-point condition cannot be used as evidence against HGT: every tree-like distance is also explained by a network with up to $h-1$ transfers, so detecting reticulation from distances alone is impossible in principle.
The explicit caterpillar construction suggests an algorithm for building such networks: for a given tree $T$, embed a caterpillar of the same height into it, solve the linear system of Theorem 5.4 for edge lengths, and leaf-graft the pendant subtrees; a natural test is to run this on random trees and check that positive solutions exist.
The root-relocation observation in Theorem 6.7 hints that the class of tree-metrizable networks may be invariant under moving the root along an arc when all pairwise distances are preserved; if so, the tree-versus-network distinction in distance data depends only on unrooted quartet structure.
Since the network distance is a convex combination of display-tree metrics, the construction also implies that the set of tree metrics is contained in the convex hull of display-tree metric sets for suitably chosen HGT networks; this convex-geometric view might connect to reconstruction algorithms based on convex mixture models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Interesting paper, but the main theorem has a proof gap that needs fixing.

The core question is worth asking: Francis and Steel showed single-reticulation HGT networks can be tree-metrizable; this paper asks how far that goes. The answer it wants is bold: any tree of height h is compatible with an HGT network with h-1 non-trivial reticulations. If that holds, then a tree metric is even weaker evidence against complex HGT histories than previously known. That is a genuinely new conceptual point, not just a parameter sweep.

What the paper does well: Lemma 3.2 gives a clean quartet criterion for three-display-tree networks, and the appendix proof is explicit. Lemma 3.4, which removes a sibling HGT arc while preserving distances, is also concretely proven with adjusted weights. The leaf-grafting Replacement Theorem (4.3) is plausible and the case breakdown is the right shape. Section 6's classification of when network-scions graft onto trees is ambitious, and the root-relocation trick for enhanced caterpillar networks is clever.

The problem is Theorem 5.4. The proof asserts that strictly positive l_j exist for the displayed linear system as "a simple exercise" and never shows it. That alone would be an omission, but the proof then claims inequalities like betaSum(T_j)gamma+(j) > betaSum(T_{j+1})l_{j+1} without deriving them from the equations; as written, the left side has a subtracted sum that can make it smaller than the right side. The equality chain in the i>b case uses a recurrence with sum_{k=1}^b betaSum(T_k)m_k, while the gamma-(j) equations give sum_{k=1}^{j-1}. That looks like an index error, and it is load-bearing because Corollary 5.5 is built entirely on Theorem 5.4. The headline result is not yet verified.

None of this is fatal to the underlying idea. The construction may well be salvageable with a proper choice of l_j and a cleaned-up recurrence. But as it stands, the central proof is incomplete.

Who this is for: mathematical phylogenetics, particularly people working on identifiability of reticulate evolution from tree-like distances. It deserves a serious referee, and I would accept it for review, but only with major revision. I would not cite it until the proof is fixed.

Referee Report

3 major / 6 minor

Summary. The paper studies horizontal gene transfer (HGT) networks whose distance metric, defined as a convex combination of distances on displayed trees, satisfies the four-point condition and is therefore a tree metric. Such networks are called tree-metrizable. The authors extend earlier work on single-reticulation networks by constructing tree-metrizable HGT networks with many non-trivial reticulation arcs. The main tools are a quartet-based characterization (Lemma 3.2), a replacement lemma for deleting an HGT arc between siblings (Lemma 3.4), a leaf-grafting theorem (Theorem 4.3), and a construction of 'caterpillar networks' claimed to be tree-metrizable on every displayed tree (Theorem 5.4). The headline consequence is Corollary 5.5: every rooted tree of height h > 2 admits a T-metrizable HGT network with h-1 non-trivial reticulation arcs. A final section studies leaf grafts with network scions and gives a partial classification result for level-2 networks.

Significance. If the main construction is correct, the paper gives a strong and somewhat surprising answer to a natural question: a tree metric alone cannot certify the absence of complex reticulate histories, because every tree metric of height h can be realized by a network with many independent HGT arcs. This substantially extends the single-reticulation examples of Francis and Steel. The paper also introduces useful conceptual tools, especially the leaf-grafting replacement theorem and the caterpillar network family. A clear strength is that Lemmas 3.2 and 3.4 have detailed appendix proofs. However, the central theorem currently rests on unproven algebraic existence assertions and on displayed formulas that do not match the accompanying case analysis; the significance is therefore conditional on a successful repair of the proof of Theorem 5.4.

major comments (3)

[Section 5, Lemma 5.3] The formulas for int(C,xaxb|xcxn) and int(C,xaxn|xbxc) are not consequences of the case analysis in the proof and are incorrect as displayed. For example, for n=6 and q={x2,x4,x5,x6} (a=2, b=4, c=5), the proof's description of internal arcs gives int(C,x2x6|x4x5) = βΣ(T2)ℓ2 + (βΣ(T1)+βΣ(T2))m2 + (βΣ(T1)+βΣ(T2)+βΣ(T3))m3, while the displayed formula gives βΣ(T2)ℓ2 + (Σ_{t=1}^{3}βΣ(Tt))m3 + (Σ_{t=1}^{4}βΣ(Tt))m4 -- a spurious m4 term and no m2 term. Similarly, for n=7, b=4, c=6, the correct coefficient of m4 is βΣ(T5)+βΣ(T6) and that of m5 is βΣ(T6), but the displayed formula gives βΣ(T5) for m4 and 2βΣ(T6) for m5. Since Theorem 5.4 explicitly invokes Lemma 5.3 to verify the hypotheses of Lemma 3.2, the main construction is not supported as written.
[Section 5, proof of Theorem 5.4] The proof asserts 'It is a simple exercise in linear algebra that there exist strictly positive values of ℓ_j for all j ≠ i that satisfy these equations' without supplying the argument. The system is not triangular: γ+(j) depends on ℓ_{j−1}, γ−(j) depends on ℓ_{j+1}, the boundary conditions set ℓ_{n−3}=ℓ_{n−2} and ℓ_2=ℓ_1, and the extra equality couples the two chains through ℓ_{i−1} and ℓ_{i+1}. Because the affine expressions have negative constant terms, positivity is a genuine constraint rather than an automatic consequence. The proof also does not address how the chosen ℓ_i is accommodated by pendant arc lengths. A constructive feasibility argument, or an explicit solution, is required before Theorem 5.4 can be accepted.
[Section 5, proof of Theorem 5.4] The proof treats the probabilities βΣ(T_j) as free positive parameters, but in the HGT network model they are not free. For a caterpillar network with independent reticulation probabilities α_i, one has βΣ(T_i)=α_i∏_{j≠i}(1−α_j) for i=1,...,n−2 and βΣ(T_{n−1})=∏_{j=1}^{n−2}(1−α_j). These relations do not fill the whole probability simplex: for n=4, the uniform distribution (1/3,1/3,1/3) is not attainable, since the formulas force α1=β1/(β1+β3), α2=β2/(β2+β3), and the resulting β1 is not 1/3. The proof never constructs the α_i or verifies that the βΣ values used in the linear system and inequalities are realizable by independent reticulation probabilities. Without this, the claimed existence of reticulation probabilities satisfying Lemma 3.2 is not established.

minor comments (6)

[Section 5, proof of Theorem 5.4] The sentence 'Fix all aj to be some arbitrary non-zero lengths' uses an undefined symbol aj; it should refer to the m_j defined just before, and the lengths should be strictly positive, not merely non-zero.
[Section 6, proof of Theorem 6.5] The proof refers to 'Lemma 6.2', but no such lemma exists; the intended reference is Theorem 6.2.
[Section 5, Corollary 5.5] The phrase 'the subtree of T induced by δ(y_i)' is not defined when δ(y_i) is a leaf; the intended construction appears to be to graft, at each vertex of the chosen length-h path, the rooted subtree attached to the off-path child, and to handle the terminal leaf separately.
[Section 3, Example 3.3] The displayed condition uses '≥' while Lemma 3.2 requires a strict inequality '>' for T1-metrizability; please correct the example to match the lemma.
[Section 4, proof of Theorem 4.3] In Case (3), the statement 'It is immediate that the inequality holds' is terse; writing the three quartet sums explicitly would make the proof easier to verify.
[General] The paper alternates between 'tree-metrized' (title and abstract) and 'tree-metrizable' (body); please standardize the terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the existence constructions are self-contained; the only concern is an unproved linear-algebra positivity step, which is a gap rather than circularity.

full rationale

The paper's main results are existence theorems: Theorem 5.4 constructs edge lengths and uses reticulation probabilities as free parameters, then verifies the four-point condition through the internal-arc criterion Lemma 3.2 (proved in Appendix A). The inequalities in the proof are checked from the recurrence equations and Lemma 5.3, not imported from the statement being proved. Corollary 5.5 combines Theorem 5.4 with the in-paper leaf-grafting Theorem 4.3. The cited work [5] provides background lemmas and the n=4 base example (Theorem 2.8), but the central construction does not reduce to that citation; no displayed equation or fitted parameter is renamed as a prediction. The proof's assertion that strictly positive l_j solving the displayed linear system exist by 'a simple exercise in linear algebra' is an omitted justification and a potential correctness gap, but it is not circular because the existence of the solution is not assumed as the conclusion. Therefore no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The new network classes (caterpillar and enhanced caterpillar networks) are explicitly defined combinatorial constructions, not hidden empirical entities, so they are not listed as invented entities. The free parameters are construction variables, not data fits; they are listed because the central existence proof chooses them by hand.

free parameters (4)

ℓ_i (HGT arc start distances) = chosen via linear system in Theorem 5.4
Distances from parent tree vertices to the starts of reticulation arcs in caterpillar networks; chosen to force internal-arc sums to satisfy Lemma 3.2.
m_i (internal edge lengths) = arbitrary positive
Internal edge lengths of the caterpillar underlying tree; the construction leaves them free ('Fix all aj to be some arbitrary non-zero lengths').
reticulation probabilities α_j (or βΣ) = arbitrary positive, summing to 1
Mixing probabilities on displayed trees; Theorem 5.4 treats βΣ as arbitrary and adjusts ℓ_i to make any target tree work.
Adjusted weights A1, A2, A3 in Lemma 3.4 = a1+αa2, a3+(1−α)a2, a5+(1−α)a4
Weights assigned to the reduced network to preserve distances when deleting a sibling HGT arc.

assumptions (4)

standard math A convex combination of metrics on X is a metric
Used to define d(N) and to ensure four-point condition checking via display trees; cited from [5, §2.3].
standard math The four-point condition characterizes tree metrics
Theorem 2.2, due to Buneman [1], is the foundation for defining tree-metrizability.
domain assumption Any tree of height h contains a caterpillar subtree on h+1 leaves as a graph embedding
Used in Corollary 5.5 to lift the caterpillar network to a T-metrizable network; standard for rooted binary trees.
ad hoc to paper Existence of strictly positive ℓ_j solving the linear system in Theorem 5.4
Asserted as 'a simple exercise in linear algebra' but not demonstrated; this carries the central caterpillar construction.

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Cite this review

Pith. "Pith review of Tree-metrizable HGT networks." pith.science (2026). https://pith.science/paper/OFUPZTUB

@misc{pith2026190808647,
  author       = {Pith},
  title        = {Pith review of: Tree-metrizable HGT networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFUPZTUB}},
  note         = {Machine review of arXiv:1908.08647}
}

read the original abstract

Phylogenetic trees are often constructed by using a metric on the set of taxa that label the leaves of the tree. While there are a number of methods for constructing a tree using a given metric, such trees will only display the metric if it satisfies the so-called "four point condition", established by Buneman in 1971. While this condition guarantees that a unique tree will display the metric, meaning that the distance between any two leaves can be found by adding the distances on arcs in the path between the leaves, it doesn't exclude the possibility that a phylogenetic network might also display the metric. This possibility was recently pointed out and "tree-metrized" networks --- that display a tree metric --- with a single reticulation were characterized. In this paper, we show that in the case of HGT (horizontal gene transfer) networks, in fact there are tree-metrized networks containing many reticulations.

Figures

Figures reproduced from arXiv: 1908.08647 by the authors.

**Figure 1.** (i) an HGT network N with HGT arcs a1, a2. Denote the other parent arcs of the reticulation vertices by a ′ 1 , a′ 2 respectively; (ii) The resulting graph after deleting a ′ 1 , a′ 2 ; (iii) The resulting display tree after deletion of unlabelled leaves and suppression of degree 2 nodes. HGT networks have the particularly useful property of having a ‘canonical’ display tree, obtained by deleting all of the reticula… view at source ↗

**Figure 2.** (i) an HGT network N, with reticulation arcs shown dashed; (ii) the underlying tree TN of N. 2.3. HGT network distances. Following [5], we define distances on a network N by treating it as a weighted union of the set of X-trees obtained by making choices at each reticulation. For each vertex v in the set VR of reticulation vertices of N, let R(v) denote the two arcs that end at v. We write Nw for a network N with w … view at source ↗

**Figure 3.** A weighted network Nw (with weights omitted) on 6 leaves. Denote the second arc ending at the same vertex as a1, a2 and a3 respectively by a ′ 1 , a′ 2 and a ′ 3 . Then by making the selection a1, a′ 2 , a3 (and thus deleting a ′ 1 , a2 and a ′ 3 ), we obtain the following display tree T w. x1 x2x3 x4 x5 x6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: The weighted display tree T w (with weights omitted) obtained from Nw in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: A network N with its three display trees, T1, T2 and T3 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: A caterpillar network on six leaves, labelled as required, except for reticulation probabilities. In order to prove that these caterpillar networks are tree-metrizable, we will have to make use of two technical lemmas. The first is Lemma 3.2, which we recall is used to…

**Figure 7.** Figure 7: The display trees T1, T2, T3 of the six-leaf caterpillar network in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Two examples of a networks formed by leaf-grafting a tree-metrizable network onto a tree. The resulting network N1 is not tree-metrizable, but the network N2 is tree-metrizable. However, if we take the network N and just move the root to leaf x4 to form N′ , and then g…

**Figure 9.** Figure 9: (i) an HGT network N with a single non-trivial biconnected component, B; (ii) The non-trivial biconnected component B of N; (iii) A minimal support network of B. We note that minimal support networks are perhaps most easily understood as the rooted equivalent of BN fr…

**Figure 10.** Figure 10: A leaf ℓ attached to the edge between two cherries. We will now define a class of networks C for which all leaf-grafts of N ∈ C onto a tree T are tree-metrizable. Definition 6.6. Let N be a network obtained by taking a caterpillar network C and adding a reticulation a…

**Figure 11.** Figure 11: (i) A caterpillar network grafted to a tree to form the network N′ in Theorem 6.7; (ii) The network N′ from (i) modified by relocating the root to the arc a. In both diagrams a triangle indicates a tree structure. Proof. Let N′′ be the network obtained by relocating t…

**Figure 12.** Figure 12: (i) The section between vertices v1, v2 and v3 in Nw; (ii) The corresponding section of Nbwb . We now consider the resulting weighted display trees. Of course, v2 and v3 may not appear in T2i−1 and T2i , but this will occur if and only if those same arcs/vertices do n…

**Figure 13.** Figure 13: (i) The relevant section of a display tree of N that contains vertices v1, v2, v3 when keeping α; (ii) The corresponding section with deletion of α. If we can set values for A1, A2, A3 so that d (Tbwbi i ,βbi) (s, t) = d(T w2i−1 2i−1 ,β2i−1) (s, t) + d(T w2i 2i ,β2i)…

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Peter Buneman, The recovery of trees from measures of dissimilarity , Mathematics in the Archaeological and Historical Sciences (1971)

work page 1971
[2]

Joseph Felsenstein, Inferring phylogenies, Sinauer Press, 2004

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Huson, New common ancestor problems in trees and directed acyclic graphs, Information Processing Letters 110 (2010), no

Johannes Fischer and Daniel H. Huson, New common ancestor problems in trees and directed acyclic graphs, Information Processing Letters 110 (2010), no. 8-9, 331–335. TREE-METRIZABLE HGT NETWORKS 23

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Huber, and Vincent Moulton, Tree-based unrooted phylogenetic networks, Bulletin of Mathematical Biology 80 (2018), no

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work page 2015
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Andrew Francis and Mike Steel, Which phylogenetic networks are merely trees with addition al arcs?, Systematic Biology 64 (2015), no. 5, 768–777

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Huson and David Bryant, Application of phylogenetic networks in evolutionary studies, Molecular Biology and Evolution 23 (2005), no

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work page 2005
[8]

Huson, Regula Rupp, and Celine Scornavacca, Phylogenetic networks: concepts, algorithms and applications , Cambridge University Press, 2010

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[9]

Hyeonsoo Jeong, Bushra Arif, Gustavo Caetano-Anoll´ es, Ky ung Mo Kim, and Arshan Nasir, Horizontal gene transfer in human-associated microorgani sms inferred by phylogenetic recon- struction and reconciliation , Scientiﬁc Reports 9 (2019), no. 1

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Carl R. Woese, On the evolution of cells , Proceedings of the National Academy of Sciences 99 (2002), no. 13, 8742–8747. Appendix A. Proofs of Lemma 3.2 and Lemma 3.4 Lemma A.1 (Lemma 3.2). Let N be a four-leaf network with leaves {x1, x2, x3, x4}. Suppose |TN | = 3,, so TN = ...

2002

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Peter Buneman, The recovery of trees from measures of dissimilarity , Mathematics in the Archaeological and Historical Sciences (1971)

work page 1971

[2] [2]

Joseph Felsenstein, Inferring phylogenies, Sinauer Press, 2004

work page 2004

[3] [3]

Huson, New common ancestor problems in trees and directed acyclic graphs, Information Processing Letters 110 (2010), no

Johannes Fischer and Daniel H. Huson, New common ancestor problems in trees and directed acyclic graphs, Information Processing Letters 110 (2010), no. 8-9, 331–335. TREE-METRIZABLE HGT NETWORKS 23

work page 2010

[4] [4]

Huber, and Vincent Moulton, Tree-based unrooted phylogenetic networks, Bulletin of Mathematical Biology 80 (2018), no

Andrew Francis, Katherina T. Huber, and Vincent Moulton, Tree-based unrooted phylogenetic networks, Bulletin of Mathematical Biology 80 (2018), no. 2, 404

work page 2018

[5] [5]

Andrew Francis and Mike Steel, Tree-like reticulation networks: When do tree-like distan ces also support reticulate evolution? , Mathematical Biosciences 259 (2015), 12–19

work page 2015

[6] [6]

5, 768–777

Andrew Francis and Mike Steel, Which phylogenetic networks are merely trees with addition al arcs?, Systematic Biology 64 (2015), no. 5, 768–777

work page 2015

[7] [7]

Huson and David Bryant, Application of phylogenetic networks in evolutionary studies, Molecular Biology and Evolution 23 (2005), no

Daniel H. Huson and David Bryant, Application of phylogenetic networks in evolutionary studies, Molecular Biology and Evolution 23 (2005), no. 2, 254–267

work page 2005

[8] [8]

Huson, Regula Rupp, and Celine Scornavacca, Phylogenetic networks: concepts, algorithms and applications , Cambridge University Press, 2010

Daniel H. Huson, Regula Rupp, and Celine Scornavacca, Phylogenetic networks: concepts, algorithms and applications , Cambridge University Press, 2010

work page 2010