## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 31, Number 3 (2017), 415-475.

### Probabilistic models for the (sub)tree(s) of life

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#### Abstract

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model species trees.

We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $\beta$-family of Aldous’ branching models.

We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered.

We then focus on the reduced tree, or coalescent tree, which is the tree spanned by species alive at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes.

We end up displaying some recent biological applications of coalescent point processes to the inference of species diversification, to conservation biology and to epidemiology.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 31, Number 3 (2017), 415-475.

**Dates**

Received: January 2016

Accepted: April 2016

First available in Project Euclid: 22 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1503388824

**Digital Object Identifier**

doi:10.1214/16-BJPS320

**Mathematical Reviews number (MathSciNet)**

MR3693976

**Zentralblatt MATH identifier**

1373.05178

**Keywords**

Random tree tree shape real tree reduced tree branching process coalescent comb phylogenetics population dynamics population genetics

#### Citation

Lambert, Amaury. Probabilistic models for the (sub)tree(s) of life. Braz. J. Probab. Stat. 31 (2017), no. 3, 415--475. doi:10.1214/16-BJPS320. https://projecteuclid.org/euclid.bjps/1503388824

#### References

- Aldous, D. (1991). The continuum random tree. I.
*The Annals of Probability***19**, 1–28. - Aldous, D. (1993). The continuum random tree. III.
*The Annals of Probability***21**, 248–289. - Aldous, D. (1996). Probability distributions on cladograms. In
*Random Discrete Structures*(A. Friedman, W. Miller, D. Aldous and R. Pemantle, eds.)**76**1–18. New York: Springer. - Aldous, D. and Popovic, L. (2005). A critical branching process model for biodiversity.
*Advances in Applied Probability***37**, 1094–1115. - Aldous, D. J. (2001). Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today.
*Statistical Science***16**, 23–34. - Barthélémy, J.-P. and Guénoche, A. (1991).
*Trees and Proximity Representations*. New York: Wiley. - Bertoin, J. (1996).
*Lévy Processes. Cambridge Tracts in Mathematics***121**. Cambridge: Cambridge Univ. Press. - Bertoin, J. (2006).
*Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics***102**. Cambridge: Cambridge Univ. Press. - Blum, M. G. and François, O. (2006). Which random processes describe the tree of life? A large-scale study of phylogenetic tree imbalance.
*Systematic Biology***55**, 685–691. - Brown, J. K. M. (1994). Probabilities of evolutionary trees.
*Systematic Biology***43**, 78–91. - Burago, D., Burago, Y. and Ivanov, S. (2001).
*A Course in Metric Geometry. Graduate Studies in Mathematics***33**. Providence, RI: American Mathematical Society. - Champagnat, N. and Lambert, A. (2012). Splitting trees with neutral Poissonian mutations I: Small families. In
*Stochastic Processes and Their Applications***122**, 1003–1033. - Champagnat, N. and Lambert, A. (2013). Splitting trees with neutral Poissonian mutations II: Largest and oldest families.
*Stochastic Processes and their Applications***123**, 1368–1414. - Delaporte, C., Achaz, G. and Lambert, A. (2016). Mutational pattern of a sample from a critical branching population.
*Journal of Mathematical Biology Journal of Mathematical Biology***73**, 627–664. - Dress, A., Moulton, V. and Terhalle, W. (1996). T-theory: An overview.
*European Journal of Combinatorics***17**, 161–175. - Duquesne, T. (2006). The coding of compact real trees by real valued functions. Preprint. Available at arXiv:math/0604106.
- Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes.
*Asterisque—Société Mathématique de France***281**. Paris: Société Mathématique de France. - Etienne, R. S., Morlon, H. and Lambert, A. (2014). Estimating the duration of speciation from phylogenies.
*Evolution***68**, 2430–2440. - Etienne, R. S. and Rosindell, J. (2012). Prolonging the past counteracts the pull of the present: Protracted speciation can explain observed slowdowns in diversification.
*Systematic Biology***61**, 204–213. - Evans, S. N. (2008).
*Probability and Real Trees. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. Lecture Notes in Mathematics***1920**. Berlin: Springer. - Evans, S. N., Pitman, J. and Winter, A. (2005). Rayleigh processes, real trees, and root growth with re-grafting.
*Probability Theory and Related Fields***134**, 81–126. - Ewens, W. J. (1972). The sampling theory of selectively neutral alleles.
*Theoretical Population Biology***3**, 87–112. Erratum*Theoretical Population Biology***3**240, 376. - Geiger, J. (1996). Size-biased and conditioned random splitting trees. In
*Stochastic Processes and Their Applications***65**, 187–207. - Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In
*Classical and Modern Branching Processes Minneapolis, MN*,*1994. IMA Vol. Math. Appl.***84**, 111–126. New York: Springer. - Haas, B. (2016). Scaling limits of Markov-branching trees and applications. Preprint. Available at arXiv:1605.07873.arXiv: 1605.07873
- Haas, B., Miermont, G., Pitman, J. and Winkel, M. (2008). Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models.
*The Annals of Probability***36**, 1790–1837.Mathematical Reviews (MathSciNet): MR2440924

Digital Object Identifier: doi:10.1214/07-AOP377

Project Euclid: euclid.aop/1221138767 - Hagen, O., Hartmann, K., Steel, M. and Stadler, T. (2015). Age-dependent speciation can explain the shape of empirical phylogenies.
*Systematic Biology***64**, 432–440. - Harding, E. F. (1971). The probabilities of rooted tree-shapes generated by random bifurcation.
*Advances in Applied Probability***3**, 44–77. - Jetz, W., Thomas, G. H., Joy, J. B., Hartmann, K. and Mooers, A. O. (2012). The global diversity of birds in space and time.
*Nature***491**, 444–448. - Kingman, J. (1982). The coalescent.
*Stochastic Processes and Their Applications***13**, 235–248. - Knuth, D. E. (1997).
*The Art of Computer Programming*. Reading, MA: Addison-Wesley. - Kyprianou, A. E. (2006).
*Introductory Lectures on Fluctuations of Lévy Processes with Applications*. Berlin: Springer. - Lambert, A. (2008). Population dynamics and random genealogies.
*Stochastic Models***24**, 45–163. - Lambert, A. (2009). The allelic partition for coalescent point processes.
*Markov Processes and Related Fields***15**, 359–386. - Lambert, A. (2010). The contour of splitting trees is a Lévy process.
*The Annals of Probability***38**, 348–395. - Lambert, A. (2011). Species abundance distributions in neutral models with immigration or mutation and general lifetimes.
*Journal of Mathematical Biology***63**, 57–72. - Lambert, A., Alexander, H. K. and Stadler, T. (2014a). Phylogenetic analysis accounting for age-dependent death and sampling with applications to epidemics.
*Journal of Theoretical Biology***352**, 60–70. - Lambert, A., Morlon, H. and Etienne, R. S. (2014b). The reconstructed tree in the lineage-based model of protracted speciation.
*Journal of Mathematical Biology***70**, 367–397.Mathematical Reviews (MathSciNet): MR3294977

Digital Object Identifier: doi:10.1007/s00285-014-0767-x - Lambert, A. and Popovic, L. (2013). The coalescent point process of branching trees.
*Annals of Applied Probability***23**, 99–144. - Lambert, A., Simatos, F. and Zwart, B. (2013). Scaling limits via excursion theory: Interplay between Crump–Mode–Jagers branching processes and processor-sharing queues.
*The Annals of Applied Probability***23**, 2357–2381. - Lambert, A. and Stadler, T. (2013). Birth–death models and coalescent point processes: The shape and probability of reconstructed phylogenies.
*Theoretical Population Biology***90**, 113–128. - Lambert, A. and Steel, M. (2013). Predicting the loss of phylogenetic diversity under non-stationary diversification models.
*Journal of Theoretical Biology***337**, 111–124. - Lambert, A. and Trapman, P. (2013). Splitting trees stopped when the first clock rings and Vervaat’s transformation.
*Journal of Applied Probability***50**, 208–227. - Lambert, A. and Uribe Bravo, G. (2016a). The comb representation of compact ultrametric spaces. Preprint. Available at arXiv:1602.08246.arXiv: 1602.08246
- Lambert, A. and Uribe Bravo, G. (2016b). Totally ordered, measured trees and splitting trees with infinite variation. Preprint. Available at arXiv:1607.02114.arXiv: 1607.02114
- Le Gall, J.-F. (1993). The uniform random tree in a Brownian excursion.
*Probability Theory and Related Fields***96**, 369–383. - Le Gall, J.-F. (2005). Random trees and applications.
*Probability Surveys***2**, 245–311. - Le Gall, J.-F. and Miermont, G. (2012). Scaling limits of random trees and planar maps. In
*Probability and Statistical Physics in Two and More Dimensions*(D. Ellwood, ed.).*Clay Math. Proc.***15**, 155–211. Providence, RI: American Mathematical Society. - Manceau, M., Lambert, A. and Morlon, H. (2015). Phylogenies support out-of-equilibrium models of biodiversity.
*Ecology Letters***18**, 347–356. - Mooers, A., Gascuel, O., Stadler, T., Li, H. and Steel, M. (2012). Branch lengths on birth–death trees and the expected loss of phylogenetic diversity.
*Systematic Biology***61**, 195–203. - Murtagh, F. (1984). Counting dendrograms: A survey.
*Discrete Applied Mathematics***7**, 191–199. - Nee, S. (2006). Birth-death models in macroevolution.
*Annual Review of Ecology, Evolution and Systematics***37**, 1–17. - Nee, S., May, R. and Harvey, P. (1994). The reconstructed evolutionary process.
*Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences***344**, 305–311. - Nee, S. and May, R. M. (1997). Extinction and the loss of evolutionary history.
*Science***278**, 692–694. - Paulin, F. (1989). The Gromov topology on $R$-trees.
*Topology and its Applications***32**, 197–221. - Pitman, J. (2006).
*Combinatorial Stochastic Processes*. In*Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002. Lecture Notes in Mathematics***1875**. Berlin: Springer. - Popovic, L. (2004). Asymptotic genealogy of a critical branching process.
*Annals of Applied Probability***14**, 2120–2148. - Richard, M. (2014). Splitting trees with neutral mutations at birth. In
*Stochastic Processes and Their Applications***124**, 3206–3230. - Semple, C. and Steel, M. A. (2003).
*Phylogenetics. Oxford Lecture Series in Mathematics and its Applications***24**. Oxford: Oxford Univ. Press. - Slowinski, J. B. (1990). Probabilities of $n$-trees under two models: A demonstration that asymmetrical interior nodes are not improbable.
*Systematic Biology***39**, 89–94. - Stadler, T. (2010). Sampling-through-time in birth–death trees.
*Journal of Theoretical Biology***267**, 396–404. - Stadler, T. (2011). Mammalian phylogeny reveals recent diversification rate shifts.
*Proceedings of the National Academy of Sciences***108**, 6187–6192. - Stanley, R. P. (1999).
*Enumerative Combinatorics. Cambridge Studies in Advanced Mathematics***62**. Cambridge: Cambridge Univ. Press. - Trapman, P. and Bootsma, M. C. J. (2009). A useful relationship between epidemiology and queueing theory: The distribution of the number of infectives at the moment of the first detection.
*Mathematical Biosciences***219**, 15–22.Mathematical Reviews (MathSciNet): MR2518217

Digital Object Identifier: doi:10.1016/j.mbs.2009.02.001

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