### Abstract

- Top of page
- Abstract
- Introduction
- Phylogenetic trees induced by speciation-extinction models
- Different classes of speciation-extinction models
- Uncovering speciation and extinction dynamics
- Uncovering speciation and extinction dynamics based on (ranked) tree shapes
- Uncovering speciation and extinction dynamics based on reconstructed trees
- Future challenges
- References

Phylogenetic trees of only extant species contain information about the underlying speciation and extinction pattern. In this review, I provide an overview over the different methodologies that recover the speciation and extinction dynamics from phylogenetic trees. Broadly, the methods can be divided into two classes: (i) methods using the phylogenetic tree shapes (i.e. trees without branch length information) allowing us to test for speciation rate variation and (ii) methods using the phylogenetic trees with branch length information allowing us to quantify speciation and extinction rates. I end the article with an overview on limitations, open questions and challenges of the reviewed methodology.

### Introduction

- Top of page
- Abstract
- Introduction
- Phylogenetic trees induced by speciation-extinction models
- Different classes of speciation-extinction models
- Uncovering speciation and extinction dynamics
- Uncovering speciation and extinction dynamics based on (ranked) tree shapes
- Uncovering speciation and extinction dynamics based on reconstructed trees
- Future challenges
- References

A phylogenetic tree represents the evolutionary relationship between species. Tips correspond to extant species and branching events correspond to speciation events (Fig. 1, middle). Classically such phylogenetic trees were reconstructed using morphological data from different species, based on the evolutionary concept that morphology of closely related species shares more characteristics than morphology of distantly related species. The increasing amount of sequence data allowed the reconstruction of many more species phylogenies, based on the evolutionary concept that genetic sequences of closely related species share more characteristics than genetic sequences of distantly related species.

Phylogenetic trees not only inform systematists about the evolutionary relationship of species, but also allow us to infer evolutionary dynamics of the considered species clade using comparative methods (Harvey & Pagel, 1991). In this review, I will focus on recent methodological developments towards understanding speciation and extinction dynamics based on phylogenetic trees. Despite rarely being termed this way, such methods are also comparative methods as the phylogenetic relationships between species are used to understand the speciation and extinction dynamics.

Inferring speciation and extinction dynamics allows us to aim at identifying the main factors that led to the biodiversity we observe today. In particular, recent methods allow us to determine the impact of the environment on speciation and extinction, as well as to identify species traits and characteristics that have a selective advantage on a macroevolutionary scale. Furthermore, the amount by which competition among species limits speciation and fosters extinction may be determined.

Speciation and extinction dynamics are inferred by fitting macroevolutionary models to phylogenetic trees, that is, by determining which model explains the data best, with sophisticated fitting methods providing a quantification for the parameters of the model. In this review article, I will first define a general model for speciation and extinction and explain how it gives rise to phylogenetic trees. Thereafter, the main body of the article will show advances in fitting such models to empirical phylogenetic trees. There exists a wide range of fitting methodology; however, all of the methods essentially aim at identifying the model which gives rise to phylogenetic trees most similar to the empirical phylogeny. I will discuss how such model fitting allows us to test macroevolutionary hypotheses as well as to quantify macroevolutionary parameters (the speciation and extinction rates). Technical challenges for the future are highlighted in little ‘Outlook’ paragraphs. The article ends with putting the methods into a broader macroevolutionary perspective, discussing current limitations and identifying main future directions necessary for improving our macroevolutionary understanding.

### Different classes of speciation-extinction models

- Top of page
- Abstract
- Introduction
- Phylogenetic trees induced by speciation-extinction models
- Different classes of speciation-extinction models
- Uncovering speciation and extinction dynamics
- Uncovering speciation and extinction dynamics based on (ranked) tree shapes
- Uncovering speciation and extinction dynamics based on reconstructed trees
- Future challenges
- References

Very general speciation and extinction dynamics can be modelled with the process introduced in the previous section. Note, though, that I assume no recombination, that is, no horizontal gene transfer or hybridization, between species. Then, the speciation-extinction models as introduced above only differ in the way how speciation and extinction rates vary through time and across species; in the most simple models, the rates are constant through time and across species. This model is termed the *constant rate birth–death model* and has been used widely as a null model (Nee *et al*., 1994b; Pybus & Harvey, 2000; Paradis, 2003). The popular Yule (1924) model is a constant rate model without extinction.

In particular, when considering large empirical clades over long evolutionary timescales, the assumption of constant rates becomes limiting, and the constant rate null model is often rejected (Pybus & Harvey, 2000; Weir, 2006; McPeek, 2008; Phillimore & Price, 2008). The models with rate variation can be divided into three classes, *species-exchangeable models*,* species-speciation-exchangeable models* and *species-non-exchangeable models*, for an overview see Table 1. The distinction of speciation-extinction models into four classes (constant rate and three nonconstant rate models) becomes important when fitting models to empirical trees. As I will discuss below, the (ranked) tree shapes under a constant rate model are exactly the same as the (ranked) tree shapes under a species-exchangeable model and a species-speciation-exchangeable model (i.e. the three classes of models give rise to the same distribution of tree shapes and ranked tree shapes), meaning we cannot distinguish between the models based on tree shapes or ranked tree shapes. I will discuss that *species-non-exchangeable models* may give rise to different ranked tree shape distributions, and such models may be identified using statistical tests based on ranked tree shapes for empirical data. I will now define the three speciation-extinction model classes allowing for rate variation in detail. I start with the most restricted model (containing the constant rate birth–death process) and generalize this class of models ending with the most general class of speciation-extinction models.

Table 1. The four classes of speciation-extinction models. The model classes become more general from top to bottom, that is, a model class contains all models from the model classes listed above. The top two model classes assume that all coexisting species are equivalent, while the species-speciation-exchangeable model only assumes exchangeability w.r.t. speciation (in particular, coexisting species may have different survival probabilities). The bottom model class further relaxes equivalence w.r.t. speciation (in particular coexisting species may have different speciation potentials). Thus, the bottom two model classes allow for modelling selection on a macroevolutionary level. Popular models within the particular model classes are provided in the middle column. The right column provides the r packages within which the particular models are available for parameter inference, based on the reconstructed trees. I want to highlight that statistical tests based on the ranked tree shape cannot distinguish between models from the top three model classes, as all these models induce a uniform distribution on ranked tree shapesModel class | Examples | Implementations |
---|

Constant rate birth–death | – | In all packages below |

Species-exchangeable | Environmental-dep. spec. & ext. Diversity-dep. spec. & ext. | H. Morlon: coal. & birth–death approx. |

Species-speciation-exchangeable | Deterministic-trait dep. ext. with asym. constant rate speciation Non-heritable-trait dep. ext. with asym. time/diversity-dep. speciation | Lambert (2010): to be implemented |

Species-non-exchangeable | Stochastic-trait dep. spec. & ext. Hidden-trait dep. spec. & ext. Deterministic-trait dep. spec. & ext. | |

First, a model is called a *species-exchangeable model* if species are exchangeable (Aldous, 2001). This means that given *k* lineages, the next event (speciation or extinction) may depend on *k* and the prior numbers of lineages through time as well as on time, but not on the particular prior complete phylogeny. Now, given the next event is a speciation event, each species is equally likely being the one speciating, and given the next event is an extinction event, each species is equally likely to go extinct. Thus, under species-exchangeable models, the rate of speciation and extinction is the same at any point in time for all species, but may change, for example, through time (*environmental dependence*), or, for example, as a function of the number of species (*diversity dependence*).

We note that this model has the following property. Label the *k* − 1 coexisting lineages prior to a speciation event with 1,…,*k* − 1 and attach the additional species with label *k*. Similarly, label the *k* coexisting lineages after an extinction event with 1,…,*k*. The discrete speciation-extinction pattern descending these *k* lineages is the complete phylogenetic tree descending the *k* lineages, ignoring the particular time interval lengths between successive events. Now, under the species-exchangeable model, the discrete speciation-extinction pattern descending a speciation event is independent of which particular coexisting species speciates (i.e. where the species *k* is attached), as future dynamics are only influenced by past overall number of lineages through time. Analog, the discrete speciation-extinction pattern descending an extinction event is independent of which particular coexisting species goes extinct at the particular extinction event.

Second, a model is called a *species-speciation-exchangeable model* if species are exchangeable w.r.t. speciation, where I define exchangeable w.r.t. speciation as (i) at any point in time given a speciation event, each species is equally likely being the one speciating, and (ii) the discrete speciation-extinction pattern descending a speciation event is independent of which species was the one speciating.

Note that species-exchangeable models are a subset of species-speciation-exchangeable models. Further note that under a species-speciation-exchangeable model, given an extinction event, some coexisting species may be more likely to go extinct than others. An example of such a model fulfilling (i) and (ii) but not belonging to the species-exchangeable models is the *asymmetric constant rate speciation and age-dependent extinction models*. Under such a model, at each speciation event, all coexisting species are equally likely to be the one speciating in an asymmetric way, and extinction may depend on the age of a species. Obviously, condition (i) is fulfilled. Furthermore, condition (ii) is fulfilled, as following a speciation event, speciation-extinction patterns only depend on the ages of the coexisting species, which are in turn independent of which species speciated. With the same argument, one can show that asymmetric time- or diversity-dependent speciation models, in which extinction depends on time, diversity and/or an arbitrary trait, and the new species of age 0 obtains a trait value independent of the mother trait (i.e. the trait is nonheritable), belong to the species-speciation-exchangeable class of models.

Third, a *species-non-exchangeable model* is a model under which some coexisting species may be more likely to speciate at a given point in time than others (i.e. condition (i) is violated), or models where all coexisting species are equally likely to speciate but the speciation-extinction pattern descending a speciation event depends on the particular species speciating (i.e. condition (ii) is violated). The *symmetric constant rate speciation and age-dependent extinction models* are an example for species-non-exchangeable models. Under such a model, at each speciation event all coexisting species are equally likely to be the one speciating in a symmetric way and extinction may depend on the age of a species. Obviously, such a symmetric probability speciation and age-dependent extinction model fulfils condition (i), but violates condition (ii) as species ages depend on the particular species speciating, and thus, future extinction patterns depend on the particular species speciating. In general, the species-non-exchangeable models can acknowledge that the speciation dynamics may depend on a trait of the species (*trait-dependence*). A trait may be, for example, species size (stochastic trait), the substitution rate in a species population (hidden trait) or the age of a species (deterministic trait). I emphasize that trait-dependent models can be used for dynamics of speciation and extinction depending on a classic trait of a species (such as size), but also depending on other characteristics of a species such as geographic location or particular ecology of a species.

The reader may wonder why I did not introduce a class of species-extinction-exchangeable models, where species are exchangeable w.r.t. extinction, but not w.r.t. speciation. I highlight below that no mathematical and methodological unique features are obvious in this model class compared with species-non-exchangeable models (which do contain the species-extinction-exchangeable models). Thus, we only discuss the species-extinction-exchangeable models indirectly when discussing species-non-exchangeable models.

#### Incomplete species sampling

In many empirical data sets, not all species of a clade are sampled. To account for this, speciation-extinction models often assume *uniform sampling*, meaning each extant species is sampled and included into the phylogeny with the same probability. Besides the uniform sampling scheme, *diversified sampling* has been suggested as a model for how biologists include species into their data sets (Höhna *et al*., 2011). Under diversified sampling, *n* of the *m* extant species are chosen such that diversity, being defined as the sum of branch lengths, is maximized. The rationale for this model is that when resources for data collection or sequencing are limited, species within a clade which are as diverse as possible, may be considered preferentially.

Clades may further be incomplete on the species level, because only one species per higher taxon (e.g. family or genus) is sampled. Such incomplete phylogenies are called higher-level phylogenies. These trees will not be focus of the present review, but see Purvis & Agapow (2002) for the shape of higher-level phylogenies, and Paradis (2003) and Alfaro *et al*. (2009) for methods estimating speciation and extinction dynamics based on such trees and Stadler & Bokma (2013) for a discussion on how higher taxon definitions may bias speciation and extinction rate estimates.

### Uncovering speciation and extinction dynamics based on (ranked) tree shapes

- Top of page
- Abstract
- Introduction
- Phylogenetic trees induced by speciation-extinction models
- Different classes of speciation-extinction models
- Uncovering speciation and extinction dynamics
- Uncovering speciation and extinction dynamics based on (ranked) tree shapes
- Uncovering speciation and extinction dynamics based on reconstructed trees
- Future challenges
- References

Tree shapes and ranked tree shapes obtained from empirical data can be used to identify patterns of different speciation and extinction dynamics in different parts of the tree. For example, accelerated speciation in one of the two clades subtending the root yields a highly *unbalanced tree*, meaning that one clade is much larger than the other. By looking at the distribution of tree shapes induced by a particular model, we can test whether an empirical tree shape is very unlikely and thus rejected under the particular model.

It has been recognized for decades that the constant rate birth–death model, that is, the simplest speciation-extinction null model, induces a uniform distribution on ranked tree shapes with *n* tips, that is, each ranked tree shape with *n* tips is equally likely (Edwards, 1970).

In this section, I want to discuss that many more models, namely all species-speciation-exchangeable models, induce a uniform distribution on ranked tree shapes with *n* tips. Thus, if we reject a constant rate birth–death model based on tree shape alone, this implies that rates of speciation must have varied across contemporaneous lineages or that the discrete speciation-extinction pattern descending a speciation event depended on the particular choice of species speciating at the speciation event. Furthermore, I will discuss that common schemes of incomplete sampling do not alter the uniform distribution on ranked tree shapes. As a consequence, hypothesis testing methods based on (ranked) tree shapes can be directly applied to incompletely sampled (ranked) tree shapes.

In the following, I will first characterize the models giving rise to a uniform distribution of ranked tree shapes and second show how to test the appropriateness of these models for given empirical (ranked) tree shapes.

#### Models inducing a uniform distribution on ranked tree shapes

First we assume that all extant species are included into the phylogenetic tree. At the end of this section, incomplete species sampling is discussed. Aldous (2001) showed that under complete species sampling, all species-exchangeable models induce the same distribution on ranked tree shapes with *n* tips, namely a uniform distribution. This result implies that even if diversification rates decline strongly through time (as observed in many studies), or accelerate near the present, tree shape itself will be unaffected provided that species are exchangeable.

Species-exchangeable models are not the only models inducing a uniform distribution on ranked tree shapes. In Lambert (2010), it is shown that also a non-exchangeable model, namely a species-speciation-exchangeable model with asymmetric constant speciation rate but age-dependent extinction rate, induces a uniform distribution on ranked tree shapes. Furthermore, in a species-speciation-exchangeable model where asymmetric speciation rate depends on time and/or number of species, and extinction rate depends on time, number of species and/or a non-heritable trait, a uniform distribution on ranked tree shapes is induced (A. Lambert & T. Stadler 2013, pers. comm.)

It turns out that all models in which species are exchangeable w.r.t. speciation (species-speciation-exchangeable models) induce a uniform distribution on ranked tree shapes. For the mathematically interested reader, I will now present a proof showing that each ranked tree shape on *n* species is equally likely under a species-speciation-exchangeable model.

Furthermore, the proof not only reveals that each ranked tree shape on *n* tips is equally likely, but also that for any fixed *n* − 1 speciation times, each ranked tree shape is equally likely, meaning that the ranked tree shape and the vector of speciation times are statistically independent. This property is very important for hypothesis testing: it means that we can perform a test based on ranked tree shape (e.g. runs statistic, see below) and a second test based on speciation times (e.g. *γ* statistic, see below) and combine the p-values of the test via multiplication, as the two tests are independent from each other.

A uniform distribution on ranked tree shapes with *n* tips induces a unique distribution on nonranked tree shapes with *n* tips: the probability of a tree shape is obtained through multiplying the probability of a ranked tree shape by the number of rankings of a possible tree shape. A formula for the number of rankings is, for example, given in Semple & Steel (2003), Corollary 2.4.3. As a consequence, all species-speciation-exchangeable models induce the same distribution on tree shape, and thus, any method performing tests based on tree shapes or ranked tree shapes cannot distinguish between species-speciation-exchangeable models. As it is often very convenient to work with uniform distributions, major focus is here given to ranked tree shapes.

Models not belonging to the species-speciation-exchangeable class can induce nonuniform ranked tree shape distributions. Violating condition (i) of the species-speciation-exchangeable definition can obviously lead to a nonuniform ranked tree shape distribution, for example, by considering a pure-birth model under which always one of the two species subtending the most recent speciation event is the one speciation next. This model yields a caterpillar tree (i.e. the tree where at each branching event, one of the two daughter clades consists of only one species) with probability 1. As mentioned above, a symmetric constant rate speciation and age-dependent extinction model is an example fulfilling condition (i) but not condition (ii) of the species-speciation-exchangeable definition. Violating (ii) can lead to a nonuniform distribution on ranked tree shapes. Consider, for example, a model under which each species dies at age 1. Symmetric speciation happens at times 0, , , and each coexisting species is equally likely to be the one speciating (i.e. condition (i) is fulfilled). All species die at time . Now, consider the probability distribution of four-species trees. By considering all possible scenarios, it turns out that the balanced (noncaterpillar) tree has probability 1, whereas the unbalanced (caterpillar) tree has probability 0. This distribution is clearly different from the uniform distribution on ranked tree shapes.

I further highlight that compared with species-speciation-exchangeable models, species-extinction-exchangeable models do not necessarily induce a uniform distribution on ranked tree shapes. Exchangeable w.r.t. extinction is defined as (i) at any point in time given an extinction event each species is equally likely being the one going extinct, and (ii) the discrete speciation-extinction pattern descending an extinction event is independent of which species was the one going extinct (i.e. for any species extinction leading to *k* lineages, we can label the *k* lineages after an extinction event with 1,…,*k*, such that their descendants follow a speciation-extinction pattern independent of which particular species went extinct). Consider for example a model under which always one of the two species subtending the most recent speciation event in the tree is the one speciating next, and all species have a constant rate of extinction. This model belongs to the species-extinction-exchangeable class, as each coexisting species is equally likely to go extinct, and future speciation-extinction pattern is independent of which species was chosen to go extinct. However, the caterpillar tree is obtained with probability 1, which is clearly different from a uniform distribution on ranked tree shapes. Thus, we do not consider species-extinction-exchangeable models as a separate class but discuss them within the species-non-exchangeable models below.

##### Incomplete species sampling

Up to here, tree shapes that are obtained by sampling all extant species were considered, and next tree shapes obtained by incomplete species sampling are considered. I will now explain that uniform and diversified sampling in species-speciation-exchangeable models yields again a uniform distribution on ranked tree shapes, that is, any hypothesis testing method based on ranked tree shapes can be applied to incomplete phylogenies (given sampling is uniform or diversified).

Consider any speciation-extinction model that induces a uniform distribution on ranked tree shapes when sampling is complete. For such a speciation-extinction model and uniform sampling, the resulting ranked tree shape distribution on *n* sampled species is again a uniform distribution and independent of the speciation times (Stadler, 2008b).

Also for such a speciation-extinction model and diversified sampling, the resulting ranked tree shape distribution given *n* sampled species is again a uniform distribution and independent of the speciation times. The reason is that under diversified sampling, we prune say *m* − *n* of a total of *m* species, such that the *m* − *n* most recent coalescent events are pruned (going backward in time). The probability of the order of the remaining coalescent events can be calculated again as in the case of complete sampling, meaning we obtain a uniform distribution on ranked tree shapes.

##### Outlook

It remains an open task to characterize all speciation-extinction models and all sampling schemes giving rise to a uniform distribution on ranked tree shapes. Only such a full characterization allows us to specify all speciation-extinction models being rejected when performing an hypothesis test based on ranked tree shapes. In particular, it is unclear whether there is a model *not* belonging to the species-speciation-exchangeable model class that gives rise to a uniform distribution on ranked tree shapes.

#### Testing evolutionary hypotheses based on (ranked) tree shapes

Tree shapes and ranked tree shapes can be used to test evolutionary hypotheses. The aim is to determine whether an empirical tree shape is very unlikely/likely under a given model, and thus the model is rejected/not rejected. Classically, a uniform distribution on ranked tree shapes with *n* tips is used as a null model, meaning all species-speciation-exchangeable models (and additional models giving rise to a uniform distribution on ranked tree shapes, if such exist) are considered as one class of models, and it is tested whether the empirical tree deviates from this class of models.

##### Unranked tree shapes

As mentioned above, phylogenetic reconstruction methods infer the tree shape, but not necessarily the ranking. Thus, the initial methods for testing evolutionary hypotheses considered the tree shape without a ranking. Guyer & Slowinski (1991) tabulated frequencies of a number of 5-taxon tree shapes. However, for larger trees, the number of tree shapes is too big to be tabulated, and so-called *summary statistics* are used to compare the tree shape distribution induced by the models to the empirical tree shape. A summary statistic summarizes the tree shape in one number (or a few numbers), widely used summary statistics as the balance of the tree (Colless index, Colless, 1982), the average path lengths from the root to a tip in the tree (Sackin index, Sackin, 1972) or the number of internal vertices having exactly two tip descendants (cherries or neighbouring pairs, Steel & Penny, 1993). The Colless and Sackin index are available within the r package apTreeshape (Bortolussi *et al*., 2009), and the cherry statistic is available within the r package ape (Paradis *et al*., 2004). An overview over tree shape statistics and their properties is, for example, given in Mooers & Heard (1997), Agapow & Purvis (2002) and Chan & Moore (2002).

In hypothesis testing, the distribution of the tree shape statistic under an hypothesized model (typically a uniform distribution on ranked tree shapes) is determined. The tree shape statistic distribution is determined analytically or via simulations, depending on the summary statistic and the hypothesized model (Blum *et al*., 2006; Rosenberg, 2006). Second, the tree shape statistic of the empirical tree is calculated. If the empirical tree shape statistic is an outlier in the model tree shape statistic distribution, the model is rejected for the considered data.

Tree shape statistics have been widely used, and a main result is that empirical trees on average tend to be less balanced than trees induced by a model with a uniform distribution on ranked tree shapes (Guyer & Slowinski, 1991; Heard, 1992; Mooers, 1995; Mooers & Heard, 1997; Blum & François, 2006).

##### Ranked tree shapes

Ranked tree shapes become available if we infer calendar time-calibrated phylogenies, but then only use the order of speciation events and ignore calendar time (e.g. due to very noisy branch length estimates). The runs statistic (Ford *et al*., 2009) can detect departure from the uniform distribution on ranked tree shapes. As the runs statistic uses ranking information in addition to tree shape information, it increases the power of evolutionary hypothesis testing compared with the classic tree shape statistics.

As the classic statistics on tree shapes, the runs statistic can be used to distinguish between models giving rise to uniform ranked tree shape distributions (i.e. all the species-speciation-exchangeable models) and models with a different ranked tree shape distribution (i.e. models belonging to the species-non-exchangeable class).

Furthermore, the runs statistic can also be employed to distinguish between models within the species-non-exchangeable model class not giving rise to a uniform distribution on ranked tree shapes. Within this class, there exists a set of models that does not induce a uniform distribution on ranked tree shapes, but given a tree shape, each ranking is equally likely. A complete characterization of such models is given in Stadler (2008a), although it remains to specify an explicit macroevolutionary interpretation. An example of a macroevolutionary model giving rise to a uniform distribution of rankings given the tree shape (but not inducing a uniform distribution on ranked tree shapes) is the constant relative probability pure-birth model (Ford *et al*., 2009), where each branching point *x* is assigned a probability with which a new species is attached to its left daughter clade compared with the right daughter clade.

In contrast, a clade within which adaptive radiations happened does not produce a uniform distribution on rankings given a tree shape. The part of the tree with the adaptive radiation has an access of internal vertices appearing sequentially in the ranking, as speciation happened unusually fast during the adaptive radiation.

The runs statistic can be used for testing for a uniform distribution on rankings given the tree shape. Thus, the runs statistic may identify adaptive radiations in different parts of the tree (or other processes giving rise to a nonuniform distribution on rankings given the tree shape). The method has been used to show that the radiation of ants during the Late Cretaceous to Early Eocene (which corresponds to the rise of angiosperms) was caused by a burst of all lineages rather than only a few lineages extant at that time (Ford *et al*., 2009).

### Future challenges

- Top of page
- Abstract
- Introduction
- Phylogenetic trees induced by speciation-extinction models
- Different classes of speciation-extinction models
- Uncovering speciation and extinction dynamics
- Uncovering speciation and extinction dynamics based on (ranked) tree shapes
- Uncovering speciation and extinction dynamics based on reconstructed trees
- Future challenges
- References

As reviewed in the previous sections, a large number of methods have been developed over the past few years to account for complex speciation and extinction dynamics.

First, methods being based on ranked tree shapes have been proposed to test the plausibility of the constant rate birth–death model (often termed equal-rates Markov model) or the even simpler Yule model. As shown here, rejection of this null model implies that all models within the species-speciation-exchangeable class are rejected simultaneously, as all these models induce a uniform distribution on ranked tree shapes.

Second, methods being based on the reconstructed tree with branch lengths have been proposed to quantify speciation and extinction rates and to test between different hypothesized speciation-extinction models (beyond only testing between species-speciation-exchangeable models and species-non-exchangeable models inducing nonuniform ranked tree shape distributions). The new methods can account for periods of declining diversity through environmental-dependent speciation and extinction rates, for the observed recent flattening of LTT plots through incomplete sampling and/or diversity-dependent rates and for the imbalance in empirical phylogenies through trait-dependent speciation rates. Closed-form solutions for the likelihood of the speciation and extinction rates given a reconstructed phylogenetic tree are known for the environmental-dependent model, while all other likelihood methods rely on numerical tools solving differential equations or inverse Laplace transforms. For an overview of available methodology, see Table 1.

I already mentioned a few rather technical challenges for the future about the specific models in the ‘Outlook’ paragraphs. Looking at the bigger picture, it will become essential to combine the approaches mentioned earlier. No clade will only be influenced by, for example, diversity-dependent diversification or only by, for example, trait-dependent diversification, but most likely by a combination of the different factors. Currently, the different hypotheses can only be tested in an exclusive way, either one hypothesis is supported or the other, or none. Although such exclusive statements may hold for the data sets considered in the studies presenting the new frameworks, most data sets will likely support a mix of factors. Formalizing one general model and inferring its parameters will reveal the importance of the different factors influencing speciation and extinction dynamics.

A general model will have limitations in inference as different speciation and extinction processes may lead to the same distribution on reconstructed phylogenetic trees, meaning that based on the reconstructed phylogenetic trees we cannot distinguish between such processes (Quental & Marshall, 2010). For example, simulations revealed that past mass extinction events leave the same fingerprint in reconstructed phylogenies as periods of stasis in diversification or adaptive radiations followed by a recent increase in diversification (Crisp & Cook, 2009; Stadler, 2011c). In such cases, sampling parameter space using Markov chain Monte Carlo (MCMC) methods may be useful to investigate the space of supported parameter combinations. Alternatively, contour plots of the likelihood surface may be considered, although this will be cumbersome as the general model has a high-dimensional parameter space. Recently, the MCMC method BayesRates (implemented in Python) for estimating time-dependent speciation and extinction rates became available (Silvestro *et al*., 2011), although the method still considers different models in an exclusive manner and tests them against each other using Bayes' factors. Investigating the full parameter space using MCMC methods allows us to identify alternative explanations for the data. To distinguish between such hypothesized explanations (e.g. mass extinction/diversification stasis), fossil data need to be considered additionally, see below.

The above methods attempt to estimate speciation and extinction rates as well as their changes. It has been widely recognized that although diversification (= speciation − extinction) can be calculated with high confidence, the turnover (extinction/speciation) is very hard to estimate: most if not all of the studies above observed this phenomenon. The diversification rate is already well informed by the final number of species as well as clade age, whereas turnover estimates require detailed information about the relative timing of speciation events. If simulated under the appropriate model, the turnover estimates seem to be accurate for large reconstructed trees though (Stadler, 2011b).

However, if we mis-specify the speciation-extinction model, it may become impossible to estimate the turnover (Rabosky, 2010). Rabosky (2010) showed that when diversification rates for each clade were drawn from a distribution of rates for simulations, but a constant rate birth–death model was fitted to the simulated data, turnover estimates were very biased. Thus, we should put increased effort into developing methods for inferring speciation and extinction rates under more complex speciation-extinction models, such that we can test the appropriateness of our models (e.g. using likelihood ratio tests) and identify aspects lacking in our models.

Developing phylogenetic methods including information from the fossil record will be a major step towards distinguishing between hypothesized models and obtaining more confined estimates about the turnover. So far, only the estimates based on the palaeontological data were compared with the estimates based on the neontological data (Morlon *et al*., 2011; Simpson *et al*., 2011). However, as the neontological data evolved from the palaeontological data, the two data sets should be considered simultaneously when inferring speciation and extinction dynamics. This could be done, for example, by using fossil counts as prior information about past biodiversity.

With the reconstruction of phylogenies including fossil tips (Ezard *et al*., 2011), we should aim for an extension of the phylogenetic methodology accounting for fossils being sampled and included in phylogenetic trees. Methodology developed for measurably evolving populations such as RNA viruses, where sampling in the past is common, can be adopted for analysing species phylogenies with fossils. In fact, in the mid-1990s, macroevolutionary phylogenetics was employed and extended to improve the understanding of viral evolution (Holmes *et al*., 1995; Nee *et al*., 1995), whereas in the future, we might do the reverse by using ideas from viral phylogenetics (Stadler, 2010; Stadler *et al*., 2012) to improve the understanding of macroevolution.

In addition to quantifying the turnover more precisely, new methods allowing us to simultaneously investigate neontological and palaeontological data sets have the potential to reveal whether the lineages leading to surviving species are significantly different from the lineages without any descendants (meaning our phylogenetic trees on extant species are a very biased sample) or whether survival of a lineage was a purely stochastic process.

All models discussed above assume certainty in species delimitation, which may not be appropriate for very recent speciation events. If we do not recognize diverging populations as separate species yet, we under-sample recent speciation events, leading to a flattened recent LTT plot, which again underestimates extinction. Etienne & Rosindell (2012) proposed a mechanism of protracted speciation that causes a flattening of the recent LTT plot. However, no likelihood equations are available yet for this model; thus, I propose to consider the phylogeny up to the time point for which we can be sure about the species delimitation and ignore the most recent past. According to Etienne & Rosindell (2012), such a time point might vary between 1 and 5 Ma depending on the organisms. The phylogenetic tree can be cut off at that time point, and the methods introduced above can be used with only a minor modification: we need to acknowledge the number of extant species at the cut-off time that went extinct before today through uniform incomplete sampling.

Last I want to point out that the discussed speciation-extinction models consider the species as a unit, although in fact species consist of a large population of individuals. Classically, we built a species tree based on genes from individuals in a population and then assume that a gene tree equals a species tree; however, this assumption does not always hold (Degnan & Rosenberg, 2009; Burbrink & Pyron, 2011). Recent methodological advances such as *Beast (Heled & Drummond, 2010) allow us to directly infer species trees based on genes from individuals; however, most species trees in the literature will in fact be gene trees. Not only the phylogenetic trees may be biased when ignoring the individuals and considering the species as a unit, but also speciation and extinction dynamics may be affected. As a species' extinction is the result of the last individual of its population dying, extinction is in fact a result of population size changes [as, for example, acknowledged in the unified neutral theory by Hubbell (2001)]. As long as population size changes within a species are quick compared with macroevolutionary dynamics, the approximation of treating a species as a unit may be very good; however, once processes happen at similar timescales, consequences of individual-based models have to be investigated, as, for example, in Davies *et al*. (2011).

I want close emphasizing that all methods for estimating macroevolutionary dynamics heavily rely on accurate phylogenetic trees. Thus, when reconstructing phylogenies based on data from extant species, appropriate methods and models together with additional data such as fossil data should be chosen carefully, and robustness of phylogenetic tree inference towards particular assumptions should be investigated. With high-quality phylogenies becoming available, and the development of general phylogenetic tools unifying dynamics at the macroevolutionary, microevolutionary and ecological scale, we have the unique opportunity to obtain a detailed understanding of the past processes and dynamics, which gave rise to the current biodiversity.