Abstract
Background
Tree reconciliation problems have long been studied in phylogenetics. A particular variant of the reconciliation problem for a gene tree T and a species tree S assumes that for each interior vertex x of T it is known whether x represents a speciation or a duplication. This problem appears in the context of analyzing orthology data.
Results
We show that S is a species tree for T if and only if S displays all rooted triples of T that have three distinct species as their leaves and are rooted in a speciation vertex. A valid reconciliation map can then be found in polynomial time. Simulated data shows that the eventlabeled gene trees convey a large amount of information on underlying species trees, even for a large percentage of losses.
Conclusions
The knowledge of event labels in a gene tree strongly constrains the possible species tree and, for a given species tree, also the possible reconciliation maps. Nevertheless, many degrees of freedom remain in the space of feasible solutions. In order to disambiguate the alternative solutions additional external constraints as well as optimization criteria could be employed.
Background
The reconstruction of the evolutionary history of a gene family is necessarily based on at least three interrelated types of information. The true phylogeny of the investigated species is required as a scaffold with which the associated gene tree must be reconcilable. Orthology or paralogy of genes found in different species determines whether an internal vertex in the gene tree corresponds to a duplication or a speciation event. Speciation events, in turn, are reflected in the species tree.
The reconciliation of gene and species trees is a widely studied problem [110]. In most practical applications, however, neither the gene tree nor the species tree can be determined unambiguously.
Although orthology information is often derived from the reconciliation of a gene tree with a species tree (cf. e.g. TreeFam [11], PhyOP [12], PHOG [13], EnsemblCompara GeneTrees [14], and MetaPhOrs [15]), recent benchmarks studies [16] have shown that orthology can also be inferred at similar levels of accuracy without the need to construct trees by means of clusteringbased approaches such as OrthoMCL [17], the algorithms underlying the COG database [18,19], InParanoid [20], or ProteinOrtho [21]. In [22] we have therefore addressed the question: how much information about the gene tree, the species tree, and their reconciliation is already contained in the orthology relation between genes?
According to Fitch's definition [23], two genes are (co)orthologous if their last common ancestor in the gene tree represents a speciation event. Otherwise, i.e., when their last common ancestor is a duplication event, they are paralogs. The orthology relation on a set of genes is therefore determined by the gene tree T and an "event labeling" that identifies each interior vertex of T as either a duplication or a speciation event. (We disregard here additional types of events such as horizontal transfer and refer to [22] for details on how such extensions might be incorporated into the mathematical framework.) One of the main results of [22], which relies on the theory of symbolic ultrametrics developed in [24], is the following: a relation on a set of genes is an orthology relation (i.e., it derives from some eventlabeled gene tree) if and only if it is a cograph (for several equivalent characterizations of cographs see [25]). Note that the cograph does not contain the full information on the eventlabeled gene tree. Instead the cograph is equivalent to the gene tree's homomorphic image obtained by collapsing adjacent events of the same type [22]. The orthology relation thus places strong and easily interpretable constraints on the gene tree.
This observation suggests that a viable approach to reconstructing histories of large gene families may start from an empirically determined orthology relation, which can be directly adjusted to conform to the requirement of being a cograph. The result is then equivalent to an (usually incompletely resolved) eventlabeled gene tree, which might be refined or used as constraint in the inference of a fully resolved gene tree. In this contribution we are concerned with the next conceptual step: the derivation of a species tree from an eventlabeled gene tree. As we shall see below, this problem is much simpler than the full tree reconciliation problem. Technically, we will approach this problem by reducing the reconciliation map from gene tree to species tree to rooted triples of genes residing in three distinct species. This is related to an approach that was developed in [26] for addressing the full tree reconciliation problem.
Methods
Definitions and notation
Phylogenetic trees
A phylogenetic tree T (on L) is a rooted tree T = (V, E), with leaf set L ⊆ V , set of directed edges E, and set of interior vertices V^{0 }= V\L that does not contain any vertices with in and outdegree one and whose root ρ_{T }∈ V has indegree zero. In order to avoid uninteresting trivial cases, we assume that L ≥ 3. The ancestor relation
It will be convenient for our discussion below to extend the ancestor relation
Rooted triples
Rooted triples are phylogenetic trees on three leaves with precisely two interior
vertices. Sometimes also called rooted triplets [28] they constitute an important concept in the context of supertree reconstruction [27,29] and will also play a major role here. Suppose L = {x, y, z}. Then we denote by ((x, y), z) the triple r with leaf set L for which the path from x to y does not intersect the path from z to the root ρ_{r }and thus, having.
Clearly, a set
The problem of determining a maximum consistent subset
The BUILD algorithm, furthermore, does not necessarily generate for a given triple
set
Event labeling, species labeling, and reconciliation map
A gene tree T arises through a series of events along a species tree S. We consider both T and S as phylogenetic trees with leaf sets L (the set of genes) and B (the set of species), respectively. We assume that L ≥ 3 and B ≥ 1. We consider only gene duplications and gene losses, which take place between speciation events, i.e., along the edges of S. Speciation events are modeled by transmitting the gene content of an ancestral lineage to each of its daughter lineages.
The true evolutionary history of a single ancestral gene thus can be thought of as
a scenario such as the one depicted in Figure 1. Since we do not consider horizontal gene transfer or lineage sorting in this contribution,
an evolutionary scenario consists of four components: (1) A true gene tree
Figure 1. Gene trees. Left: Example of an evolutionary scenario showing the evolution of a gene family. The corresponding
true gene tree
In order to allow
The true gene tree
Furthermore, we can observe a map σ: L → B that assigns to each extant gene the species in which it resides. Of course, for x ∈ L we have
The observable part of the species tree S = (W H) is the restriction
The evolutionary scenario also implies an event labeling map
(C) Let z ∈ V be a speciation vertex, i.e., t(z) = ·, and let T' and T" be subtrees of T rooted in two distinct children of z. Then σ (T') ∩ σ (T") = ∅.
Note the we do not require the converse, i.e., from the disjointness of the species sets σ (T') and σ(T") we do not conclude that their last common ancestor is a speciation vertex.
For x, y ∈ L and z = lca_{T }(x, y) it immediately follows from condition (C) that if t(lca_{T }(x, y)) = • then σ(x) ≠ σ(y) since, by assumption, x and y are leaves in distinct subtrees below z. Equivalently, two distinct genes x ≠ y in L for which σ(x) = σ(y) holds, that is, they are contained in the same species of B, must have originated from a duplication event, i.e., t(lca_{T }(x, y)) = □. Thus we can regard σ as a proper vertex coloring of the cograph corresponding to (T, t).
Let us now consider the properties of the restriction of
Definition 1. Suppose that B is a set of species, that S = (W, H) is a phylogenetic tree on B, that T = (V, E) is a gene tree with leaf set L and that σ : L → B and
(i) If
(ii) If t(x) = • then µ (x) ∈W \ B.
(iii) If t(x) = □ then µ(x) ∈ H.
(iv) Let x, y ∈ V with
1. If t(x) = t(y) = □ then
2. If t(x) = t(y) = • or t(x) ≠ t(y) then
(v) If t(x) = • then µ(x) = lca_{S}(σ(L(x)))
We call µ the reconciliation map from (T,t, σ ) to S.
We note that µ^{1}(ρ_{S}) = ∅ holds as an immediate consequence of property (v), which implies that no speciation node can be mapped above lca_{S}(B), the unique child of ρ_{S}.
We illustrate this definition by means of an example in Figure 2 and remark that it is consistent with the definition of reconciliation maps for the case when the event labeling t on T is not known [38]. Continuing with our notation from Definition 1 for the remainder of this section, we easily derive their axiom set as
Figure 2. Mapping μ. Example of the mapping μ of nodes of the gene tree T to the species tree S. Speciation nodes in the gene tree (red circles) are mapped to nodes in the species tree, duplication nodes (blue squares) are mapped to edges in the species tree. σ is shown as dashed green arrows. For clarity of exposition, we have identified the leaves of the gene tree on the left with the species they reside in via the map σ.
Lemma 2. If µ is a reconciliation map from (T,t, σ) to S and L is the leaf set of T then, for all x ∈ V.
(D1) x ∈ L implies µ (x) = σ (x).
(D2.a) µ(x) ∈ W implies µ (x) = lca_{S}(σ (L(x))).
(D2.b) µ (x) ∈ H implies
(D3) Suppose x, y ∈ V such that
Proof. Suppose x ∈ V. Then (D1) is equivalent to (i) and the fact that
For T a gene tree, B a set of species and maps σ and t as above, our goal is now to characterize (1) those (T,t, σ) for which a species tree on B exists and (2) species trees on B that are species trees for (T,t, σ).
Results and discussion
Results
Unless stated otherwise, we continue with our assumptions on B, (T,t, σ), and S as stated in Definition 1. We start with the simple observation that a reconciliation map from (T,t, σ) to S preserves the ancestor order of T and hence T imposes a strong constraint on the relationship of most recent common ancestors in S:
Lemma 3. Let µ : V → W ∪ H be a reconciliation map from (T,t, σ) to S. Then
holds for all x, y ∈ V.
Proof. Assume that x and y are distinct vertices of T. Consider the unique path P connecting x with y. P is uniquely subdivided into a path P' from x to lca_{T }(x, y) and a path P" from lca_{T }(x, y) to y. Condition (iv) implies that the images of the vertices of P' and P" under µ, resp., are ordered in S with regards to
Equation (1) is well known to hold for gene tree/species reconciliation in the absence of a prescribed event labeling in T.
Since a phylogenetic tree (in the original sense) T is uniquely determined by its induced triple set
As we shall see below,
Lemma 4. If µ is a reconciliation map from (T,t, σ) to S and
Proof. Put
Now suppose that t(lca_{T }(x, y)) = □ and therefore, µ (lca_{T }(x, y)) ∈ H. Moreover, µ (lca_{T }(x, y, z)) ∈ W holds. Hence, Lemma 3 and property (iv) together imply that
It is important to note that a similar argument cannot be made for triples in
Figure 3. Triples with duplication event at the root. Triples from T whose root is a duplication event are in general not displayed from the species tree S. (a) Triple with duplication event at the root obtained from the true evolutionary history of T shown in panel (b). Panel (c) is the true species tree. In the triple (a) the species y appears as the outgroup even though the x is the outgroup in the true species tree.
Definition 5. For (T,t, σ), we define the set
As an immediate consequence of Lemma 4,
Theorem 6. Let S be a species tree with leaf set B. Then there exists a reconciliation map µ
from (T,t, σ) to S whenever S displays all triples in
Proof. Recall that L is the leaf set of T = (V, E). Put S = (W, H) and
We explicitly construct the map µ : G → W as follows. For all x ∈ V , we put
(M2) µ(x) = lca_{S}(σ(L(x))) if t(x) = •.
Note that alternative (M1) ensures that µ satisfies Condition (i). Also note that in view of the simple consequence following the statement of Condition (C) we have for all x ∈ V with t(x) = • that there are leaves y', y" ∈ L(x) with σ(y') ≠ σ(y"). Thus lca_{S}(µ(L(x)) ∈ W \ B, i.e. µ satisfies Condition (ii). Also note that, by definition, alternative (M2) ensures that µ satisfies Condition (v).
Claim: If x, y ∈ G with
Since y cannot be a leaf of T as
Now suppose t(x) = •. Again by the simple consequence following Condition (C), there are leaves x', x" ∈ L(x) with a = σ(x') ≠ σ(x") = b. Since
Next, we extend the map µ to the entire vertex set V of T using the following observation. Let x ∈ V with t(x) = □. We know by Lemma 3 that µ(x) is an edge [u, v] ∈ H so that
(M3) µ(x) = [u, lca_{S}(σ(L(x)))] if t(x) = □.
which now makes μ a map from V to W ∪ H.
By construction, Conditions (iii), (iv.2) and (v) are thus satisfied by μ. On the other hand, if there is a speciation vertex y between two duplication vertices x and x' of T , i.e.,
It follows that μ is a reconciliation map from (T,t, σ) to S. □
Corollary 7. Suppose that S is a species tree for (T,t, σ) and that L and B are the leaf sets of T and S, respectively. Then a reconciliation map μ from (T,t, σ) to S can be constructed in O(LB).
Proof. In order to find the image of an interior vertex x of T under μ, it suffices to determine σ (L(x)) (which can be done for all x simultaneously, e.g. by bottom up transversal of T in O(LB) time) and lca_{S}(σ(L(x))). The latter task can be solved in linear time using the idea presented in [39] to calculate the lowest common ancestor for a group of nodes in the species tree. □
We remark that given a species tree S on B that displays all triples in
Lemma 4 implies that consistency of the triple set
Theorem 8. There is a species tree on B for (T,t, σ) if and only if the triple set
We remark that a related result is proven in [26, Theorem.5] for the full tree reconciliation problem starting from a forest of gene trees.
It may be surprising that there are no strong restrictions on the set
Theorem 9. For every set
Proof. Irrespective of whether
We remark that the gene tree constructed in the proof of Theorem 9 can be made into
a binary tree by splitting the root ρ_{T }into a series of duplication and loss events so that each subtree is the descendant
of a different paralog. Since by Theorem. 9 there are no restrictions on the possible
triple sets
Figure 4. Inferred species trees. The set
Results for simulated gene trees
In order to determine empirically how much information on the species tree we can
hope to find in event labeled gene trees, we simulated species trees together with
corresponding eventlabeled gene trees with different duplication and loss rates.
Approximately 150 species trees with 10 to 100 species were generated according to
the "age model" [40]. These trees are balanced and the edge lengths are normalized so that the total length
of the path from the root to each leaf is 1. For each species tree, we then simulated
a gene tree as described in [41], with duplication and loss rate parameters r ∈ 0[1] sampled uniformly. Events are modeled by a Poisson distribution with parameter r · ℓ, where ℓ is the length of an edge as generated by the age model. Losses were additionally constrained
to retain at least one copy in each species, i.e., σ(L) = B is enforced. After determining the triple set
The results are summarized in Figure 5. Not surprisingly, the recoverable information decreases in particular with the rate of gene loss. Nevertheless, at least 50% of the splits in the species tree are recoverable even at very high loss rates. For moderate loss rates, in particular when gene losses are less frequent than gene duplications, nearly the complete information on the species tree is preserved. It is interesting to note that BUILD does not incorporate splits that are not present in the input tree, although this is not mathematically guaranteed.
Figure 5. Recovered splits in species trees. Left: Heat map that represents the percentage of recovered splits in the inferred species tree from triples obtained from simulated eventlabeled gene trees with different loss and duplication rates. Right: Scattergram that shows the average of losses and duplications in the generated data and the accuracy of the inferred species tree.
Discussion
Eventlabeled gene trees can be obtained by combining the reconstruction of gene phylogenies with methods for orthology detection. Orthology alone already encapsulates partial information on the gene tree. More precisely, the orthology relation is equivalent to a homomorphic image of the gene tree in which adjacent vertices denote different types of events. We discussed here the properties of reconciliation maps μ from a gene tree T along with an event labelling map t and a gene to species assignment map σ to a species tree S. We show that (T,t) event labeled gene trees for which a species tree exists can be characterized in terms of the set σ of triples that is easily constructed from a subset of triples of T. Simulated data shows, furthermore, that such trees convey a large amount of information on the underlying species tree, even if the gene loss rate is high.
It can be expected that for reallife data the tree T contains errors so that
For a given species tree S, it is rather easy to find a reconciliation map μ from (T,t, σ) to S. A simple solution μ is closely related to the socalled LCA reconcilation: every node x of T is mapped to the last common ancestor of the species below it, lca_{S}(σ(L(x))) or to the edge immediately above it, depending on whether x is speciation or a duplication node. While this solution is unique for the speciation nodes, alternative mappings are possible for the duplication nodes. The set of possible reconciliation maps can still be very large despite the specified event labels. If the event labeling t is unknown, there is a reconciliation from any gene tree T to any species tree S, realized in particular by the LCA reconciliation, see e.g. [26,38]. The reconciliation then defines the event types. Typically, a parsimony rule is then employed to choose a reconciliation map in which the number of duplications and losses is minimized, see e.g. [1,4,5,9]. In our setting, on the other hand, the event types are prescribed. This restricts the possible reconciliation maps so that the gene tree cannot be reconciled with an arbitrary species tree any more. Since the observable events on the gene tree are fixed, the possible reconciliations cannot differ in the number of duplications. Still, one may be interested in reconciliation maps that minimize the number of loss events. An alternative is to maximize the number of duplication events that map to the same edge in S to account for whole genome and chromosomal duplication events [9].
Conclusions
Our approach to the reconciliation problem via eventlabeled gene trees opens up some interesting new avenues to understanding orthology. In particular, the results in this contribution combined with those in [22] concerning cographs should ultimately lead to a method for automatically generating orthology relations that takes into account species relationships without having to explicitly compute gene trees. This is potentially very useful since gene tree estimation is one of the weak points of most current approaches to orthology analysis.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed to the development of the theory. MHR and NW produced the simulated data. All authors contributed to writing, reading, and approving the final manuscript.
Acknowledgements
This work was supported in part by the the Volkswagen Stiftung (proj. no. I/82719) and the Deutsche Forschungsgemeinschaft (SPP1174 "Deep Metazoan Phylogeny", proj. nos. STA 850/2 and STA 850/3).
This article has been published as part of BMC Bioinformatics Volume 13 Supplement 19, 2012: Proceedings of the Tenth Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/13/S19
References

Guigó R, Muchnik I, Smith TF: Reconstruction of ancient molecular phylogeny.
Mol Phylogenet Evol 1996, 6:189213. PubMed Abstract  Publisher Full Text

Page RD, Charleston MA: From gene to organismal phylogeny: reconciled trees and the gene tree/species tree problem.
Mol Phylogenet Evol 1997, 7:231240. PubMed Abstract  Publisher Full Text

Arvestad L, Berglund AC, Lagergren J, Sennblad B: Bayesian gene/species tree reconciliation and orthology analysis using MCMC.
Bioinformatics 2003, 19:i7i15. PubMed Abstract  Publisher Full Text

Bonizzoni P, Della Vedova G, Dondi R: Reconciling a gene tree to a species tree under the duplication cost model.
Theor Comp Sci 2005, 347:3653. Publisher Full Text

Górecki P, J T: DSLtrees: A model of evolutionary scenarios.
Theor Comp Sci 2006, 359:378399. Publisher Full Text

Hahn MW: Bias in phylogenetic tree reconciliation methods: implications for vertebrate genome evolution.
Genome Biol 2007, 8:R141. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Bansal MS, Eulenstein O: The multiple gene duplication problem revisited.
Bioinformatics 2008, 24:i132i138. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Chauve C, Doyon JP, ElMabrouk N: Gene family evolution by duplication, speciation, and loss.
J Comput Biol 2008, 15:10431062. PubMed Abstract  Publisher Full Text

Burleigh JG, Bansal MS, Wehe A, Eulenstein O: Locating largescale gene duplication events through reconciled trees: implications for identifying ancient polyploidy events in plants.
J Comput Biol 2009, 16:10711083. PubMed Abstract  Publisher Full Text

Larget BR, Kotha SK, Dewey CN, Ane C: BUCKy: gene tree/species tree reconciliation with Bayesian concordance analysis.
Bioinformatics 2010, 26:29102911. PubMed Abstract  Publisher Full Text

Li H, Coghlan A, Ruan J, Coin LJ, Hériché JK, Osmotherly L, Li R, Liu T, Zhang Z, Bolund L, Wong GK, Zheng W, Dehal P, Wang J, Durbin R: TreeFam: a curated database of phylogenetic trees of animal gene families.
Nucleic Acids Res 2006, 34:D572D580. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Goodstadt L, Ponting CP: Phylogenetic reconstruction of orthology, paralogy, and conserved synteny for dog and human.
PLoS Comput Biol 2006, 2:e133. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Datta RS, Meacham C, Samad B, Neyer C, Sjölander K: Berkeley PHOG: PhyloFacts orthology group prediction web server.
Nucl Acids Res 2009, 37:W84W89. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Hubbard TJ, Aken BL, Beal K, Ballester B, Caccamo M, Chen Y, Clarke L, Coates G, Cunningham F, Cutts T, Down T, Dyer SC, Fitzgerald S, FernandezBanet J, Graf S, Haider S, Hammond M, Herrero J, Holland R, Howe K, Howe K, Johnson N, Kahari A, Keefe D, Kokocinski F, Kulesha E, Lawson D, Longden I, Melsopp C, Megy K, Meidl P, Ouverdin B, Parker A, Prlic A, Rice S, Rios D, Schuster M, Sealy I, Severin J, Slater G, Smedley D, Spudich G, Trevanion S, Vilella A, Vogel J, White S, Wood M, Cox T, Curwen V, Durbin R, FernandezSuarez XM, Flicek P, Kasprzyk A, Proctor G, Searle S, Smith J, UretaVidal A, Birney E: Ensembl 2007.
Nucleic Acids Res 2007, 35:D610617. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Pryszcz LP, HuertaCepas J, Gabaldón T: MetaPhOrs: orthology and paralogy predictions from multiple phylogenetic evidence using a consistencybased confidence score.
Nucleic Acids Res 2011, 39:e32. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Altenhoff AM, Dessimoz C: Phylogenetic and functional assessment of orthologs inference projects and methods.
PLoS Comput Biol 2009, 5:e1000262. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Li L, Stoeckert CJ Jr, Roos DS: OrthoMCL: identification of ortholog groups for eukaryotic genomes.
Genome Res 2003, 13:21782189. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Tatusov RL, Galperin MY, Natale DA, Koonin EV: The COG database: a tool for genomescale analysis of protein functions and evolution.
Nucleic Acids Res 2000, 28:3336. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Wheeler DL, Barrett T, Benson DA, Bryant SH, Canese K, Chetvernin V, Church DM, Dicuccio M, Edgar R, Federhen S, Feolo M, Geer LY, Helmberg W, Kapustin Y, Khovayko O, Landsman D, Lipman DJ, Madden TL, Maglott DR, Miller V, Ostell J, Pruitt KD, Schuler GD, Shumway M, Sequeira E, Sherry ST, Sirotkin K, Souvorov A, Starchenko G, Tatusov RL, Tatusova TA, Wagner L, Yaschenko E: Database resources of the National Center for Biotechnology Information.
Nucleic Acids Res 2008, 36:D13D21. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Berglund AC, Sjölund E, Ostlund G, Sonnhammer EL: InParanoid 6: eukaryotic ortholog clusters with inparalogs.
Nucleic Acids Res 2008, 36:D263D266. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Lechner M, Findeiß S, Steiner L, Marz M, Stadler PF, Prohaska SJ: Proteinortho: Detection of (Co)Orthologs in LargeScale Analysis.
BMC Bioinformatics 2011, 12:124. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Hellmuth M, HernandezRosales M, Huber KT, Moulton V, Stadler PF, Wieseke N: Orthology Relations, Symbolic Ultrametrics, and Cographs.
J Math Biol 2012. PubMed Abstract  Publisher Full Text

Fitch WM: Homology: a personal view on some of the problems.
Trends Genet 2000, 16:227231. PubMed Abstract  Publisher Full Text

Böcker S, Dress AWM: Recovering symbolically dated, rooted trees from symbolic ultrametrics.
Adv Math 1998, 138:105125. Publisher Full Text

Brandstädt A, Le VB, Spinrad JP: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications, Philadephia: Soc. Ind. Appl. Math; 1999.

Chauve C, ElMabrouk N: New Perspectives on Gene Family Evolution: Losses in Reconciliation and a Link with Supertrees.

Semple C, Steel M: Phylogenetics, Volume 24 of Oxford Lecture Series in Mathematics and its Applications. Oxford, UK: Oxford University Press; 2003.

Dress AWM, Huber KT, Koolen J, Moulton V, Spillner A: Basic Phylogenetic Combinatorics. Cambridge: Cambridge University Press; 2011.

BinindaEmonds O: Phylogenetic Supertrees. Dordrecht, NL: Kluwer Academic Press; 2004.

Aho AV, Sagiv Y, Szymanski TG, Ullman JD: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions.
SIAM J Comput 1981, 10:405421. Publisher Full Text

Rauch Henzinger M, King V, Warnow T: Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology.
Algorithmica 1999, 24:113. Publisher Full Text

Jansson J, Ng JHK, Sadakane K, Sung WK: Rooted maximum agreement supertrees.
Algorithmica 2005, 43:293307. Publisher Full Text

Byrka J, Gawrychowski P, Huber KT, Kelk S: Worstcase optimal approximation algorithms for maximizing triplet consistency within phylogenetic networks.
J Discr Alg 2010, 8:6575. Publisher Full Text

van Iersel L, Kelk S, Mnich M: Uniqueness, intractability and exact algorithms: reflections on leve lk phylogenetic networks.
J Bioinf Comp Biol 2009, 7:597623. PubMed Abstract  Publisher Full Text

Byrka J, Guillemot S, Jansson J: New results on optimizing rooted triplets consistency.
Discr Appl Math 2010, 158:11361147. Publisher Full Text

Semple C: Reconstructing minimal rooted trees.
Discr Appl Math 2003, 127:489503. Publisher Full Text

Jansson J, Lemence RS, Lingas A: The Complexity of Inferring a Minimally Resolved Phylogenetic Supertree.
SIAM J Comput 2012, 41:272291. Publisher Full Text

Doyon JP, Chauve C, Hamel S: Space of Gene/Species Trees Reconciliations and Parsimonious Models.
J Comp Biol 2009, 16:13991418. PubMed Abstract  Publisher Full Text

Zhang L: On a MirkinMuchnikSmith conjecture for comparing molecular phylogenies.
J Comput Biol 1997, 4:177187. PubMed Abstract  Publisher Full Text

KellerSchmidt S, Tuğrul M, Eguíluz VM, HernándezGarcíi E, Klemm K: An Age Dependent Branching Model for Macroevolution.
Tech Rep 2010.
1012.3298v1, arXiv

HernandezRosales M, Wieseke N, Hellmuth M, Stadler PF: Simulation of Gene Family Histories. [http://www.bioinf.unileipzig.de/Publications/PREPRINTS/12017.pdf] webcite

Aho AV, Sagiv Y, Szymanski TG, Ullman JD: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions.
SIAM J Comput 1981, 10:405421. Publisher Full Text

Jansson J: On the Complexity of Inferring Rooted Evolutionary Trees.

Wu BY: Constructing the Maximum Consensus Tree from Rooted Triples.

He YJ, Huynh TN, Jansson J, Sung WK: Inferring phylogenetic relationships avoiding forbidden rooted triplets.
J Bioinform Comput Biol 2006, 4:5974. PubMed Abstract  Publisher Full Text

Ng MP, Wormald NC: Reconstruction of rooted trees from subtrees.
Discr Appl Math 1996, 69:1931. Publisher Full Text

Constantinescu M, Sankoff D: An efficient algorithm for supertrees.
J Classification 1995, 12:101112. Publisher Full Text

Bryant D, Steel M: Extension Operations on Sets of LeafLabeled Trees.
Adv Appl Math 1995, 16:425453. Publisher Full Text