R. Abraham and J. Delmas, Williams??? decomposition of the L??vy continuum random tree and simultaneous extinction probability for populations with neutral mutations, Stochastic Processes and their Applications, pp.1124-1143, 2009.
DOI : 10.1016/j.spa.2008.06.001

D. Aldous, The Continuum Random Tree. I, The Annals of Probability, vol.19, issue.1, pp.1-28, 1991.
DOI : 10.1214/aop/1176990534

D. Aldous, The Continuum Random Tree. I, The Annals of Probability, vol.19, issue.1, pp.248-289, 1993.
DOI : 10.1214/aop/1176990534

K. B. Athreya and P. E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, 0196.
DOI : 10.1007/978-3-642-65371-1

D. Aldous and L. Popovic, A critical branching process model for biodiversity, Advances in Applied Probability, vol.5, issue.04, pp.1094-1115, 2005.
DOI : 10.1006/jtbi.2000.2032

URL : http://arxiv.org/abs/math/0410402

G. Alsmeyer and U. Rösler, Asexual Versus Promiscuous Bisexual Galton-Watson Processes: The Extinction Probability Ratio, The Annals of Applied Probability, vol.12, issue.1, pp.125-142, 2002.
DOI : 10.1214/aoap/1015961158

H. Bi and J. Delmas, Total length of the genealogical tree for quadratic stationary continuous-state branching processes, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.52, issue.3
DOI : 10.1214/15-AIHP683

URL : https://hal.archives-ouvertes.fr/hal-01024986

N. Becker, On parametric estimation for mortal branching processes, Biometrika, vol.61, issue.2, pp.393-399, 1974.
DOI : 10.1093/biomet/61.2.393

N. Becker, Estimation for Discrete Time Branching Processes with Application to Epidemics, Biometrics, vol.33, issue.3, pp.515-522, 1977.
DOI : 10.2307/2529366

J. Bertoin, An Extension of Pitman's Theorem for Spectrally Positive Levy Processes, The Annals of Probability, vol.20, issue.3, pp.1464-1483, 1992.
DOI : 10.1214/aop/1176989701

J. Bertoin, Splitting at the infimum and excursions in half-lines for random walks and L??vy processes, Stochastic Processes and their Applications, vol.47, issue.1, pp.17-35, 1993.
DOI : 10.1016/0304-4149(93)90092-I

J. Bertoin, Lévy processes, volume 121 of Cambridge Tracts in Mathematics, 1996.

J. Bertoin, Exponential decay and ergodicity of completely asymmetric L??vy processes in a finite interval, The Annals of Applied Probability, vol.7, issue.1, pp.156-169, 1997.
DOI : 10.1214/aoap/1034625257

M. Ba, E. Pardoux, and A. Sow, Binary Trees, Exploration Processes, and an Extended Ray-Knight Theorem, Journal of Applied Probability, vol.281, issue.01, pp.210-225, 2012.
DOI : 10.1002/cpa.3160240206

URL : http://projecteuclid.org/download/pdfview_1/euclid.jap/1331216843

L. Chaumont and R. A. Doney, On L??vy processes conditioned to stay positive., Electronic Journal of Probability, vol.10, issue.0, pp.948-961, 2005.
DOI : 10.1214/EJP.v10-261

L. Chaumont, Sur certains processus de l??vy conditionn??s ?? rester positifs, Stochastics An International Journal of Probability and Stochastic Processes, vol.47, issue.1, pp.1-20, 1994.
DOI : 10.1080/17442509408833880

L. Chaumont, Conditionings and path decompositions for Lévy processes. Stochastic Process, Appl, vol.64, issue.1, pp.39-54, 1996.
DOI : 10.1016/s0304-4149(96)00081-6

URL : http://doi.org/10.1016/s0304-4149(96)00081-6

L. Chaumont, On the law of the supremum of L??vy processes, The Annals of Probability, vol.41, issue.3A, pp.1191-1217, 2013.
DOI : 10.1214/11-AOP708

E. Ma, A. Caballero, G. Lambert, and . Bravo, Proof(s) of the lamperti representation of continuous-state branching processes, Probab. Surveys, vol.6, issue.0, pp.62-89, 2009.

M. Dávila, F. , and A. Lambert, Time reversal dualities for some random forests, ALEA Lat. Am. J. Probab. Math. Stat, vol.12, issue.1, pp.399-426, 2015.

M. Dávila, F. , and A. Lambert, Branching processes seen from their extinction time via path decompositions of reflected Lévy processes. ArXiv e-prints, 1610.

J. Delmas and O. Hénard, A Williams decomposition for spatially dependent superprocesses, Electronic Journal of Probability, vol.18, issue.0, p.2013
DOI : 10.1214/EJP.v18-1801

URL : https://hal.archives-ouvertes.fr/hal-00601539

T. Duquesne and J. Gall, Random trees, Lévy processes and spatial branching processes, Astérisque, issue.281, p.147, 2002.

R. A. Doney, Fluctuation Theory for L??vy Processes, Lecture Notes in Mathematics, vol.1897, 2007.
DOI : 10.1007/978-1-4612-0197-7_3

A. Drummond, O. G. Pybus, and A. Rambaut, Inference of Viral Evolutionary Rates from Molecular Sequences, Adv Parasitol, vol.54, pp.331-358, 2003.
DOI : 10.1016/S0065-308X(03)54008-8

T. Duquesne, Path decompositions for real Levy processes, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.39, issue.2, pp.339-370, 2003.
DOI : 10.1016/S0246-0203(02)00004-3

URL : http://arxiv.org/abs/math/0509520

S. Rampal, B. Etienne, T. Haegeman, T. Stadler, P. N. Aze et al., Diversity-dependence brings molecular phylogenies closer to agreement with the fossil record, Proceedings of the Royal Society B: Biological Sciences, 2011.

W. Warren and . Esty, The reverse Galton-Watson process, J. Appl. Probability, vol.12, issue.3, pp.574-580, 1975.

D. W. Simon, O. G. Frost, J. R. Pybus, C. Gog, S. Viboud et al., Eight challenges in phylodynamic inference, Challenges in Modelling Infectious {DIsease} Dynamics, pp.88-92, 2015.

J. Geiger, Size-biased and conditioned random splitting trees. Stochastic Process, Appl, vol.65, issue.2, pp.187-207, 1996.
DOI : 10.1016/s0304-4149(96)00108-1

URL : http://doi.org/10.1016/s0304-4149(96)00108-1

J. Geiger and G. Kersting, Depth???First Search of Random Trees, and Poisson Point Processes, Classical and modern branching processes, pp.111-126, 1994.
DOI : 10.1007/978-1-4612-1862-3_8

P. Greenwood and J. Pitman, Fluctuation identities for l??vy processes and splitting at the maximum, Advances in Applied Probability, vol.XIV, issue.04, pp.893-902, 1980.
DOI : 10.1090/S0002-9947-1977-0433606-6

T. Bryan, O. G. Grenfell, J. R. Pybus, J. L. Gog, J. M. Wood et al., Unifying the epidemiological and evolutionary dynamics of pathogens, Science, issue.5656, pp.303327-332, 2004.

P. Jagers, Branching processes with biological applications, Wiley Series in Probability and Mathematical Statistics?Applied Probability and Statistics, 1975.

P. Jagers, The Growth and Stabilization of Populations, Statistical Science, vol.6, issue.3, pp.269-274, 1991.
DOI : 10.1214/ss/1177011694

J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 2003.
DOI : 10.1007/978-3-662-02514-7

M. Kimmel and D. E. Axelrod, Branching processes in biology, Interdisciplinary Applied Mathematics, vol.19, 2002.
DOI : 10.1007/978-1-4939-1559-0

M. Kac, Discrete thoughts : essays on mathematics, science, and philosophy, 2008.

G. David and . Kendall, Stochastic processes and population growth, Journal of the Royal Statistical Society. Series B (Methodological), vol.11, issue.2, pp.230-282, 1949.

C. Fima, U. Klebaner, S. Rösler, and . Sagitov, Transformations of galtonwatson processes and linear fractional reproduction, Advances in Applied Probability, vol.39, issue.4, pp.1036-1053, 2007.

A. E. Kyprianou, Introductory lectures on fluctuations of Lévy processes with applications. Universitext, 2006.

J. Lamperti, Continuous state branching processes, Bulletin of the American Mathematical Society, vol.73, issue.3, pp.382-386, 1967.
DOI : 10.1090/S0002-9904-1967-11762-2

A. Lambert, The contour of splitting trees is a L??vy process, The Annals of Probability, vol.38, issue.1, pp.348-395, 2010.
DOI : 10.1214/09-AOP485

A. Lambert, Species abundance distributions in neutral models with immigration or mutation and general lifetimes, Journal of Mathematical Biology, vol.6, issue.1, pp.57-72, 2011.
DOI : 10.1007/s00285-010-0361-9

URL : https://hal.archives-ouvertes.fr/hal-00607759

A. Lambert, H. K. Alexander, and T. Stadler, Phylogenetic analysis accounting for age-dependent death and sampling with applications to epidemics, Journal of Theoretical Biology, vol.352, issue.0, pp.60-70, 2014.
DOI : 10.1016/j.jtbi.2014.02.031

URL : https://hal.archives-ouvertes.fr/hal-00834706

A. Lambert and G. Bravo, Totally ordered measured trees and splitting trees with infinite variation. ArXiv e-prints, 1607.

J. Gall, Random trees and applications, Probability Surveys, vol.2, issue.0, pp.245-311, 2005.
DOI : 10.1214/154957805100000140

E. Gabriel, H. F. Leventhal, S. Günthard, T. Bonhoeffer, and . Stadler, Using an epidemiological model for phylogenetic inference reveals density-dependence in hiv transmission, Molecular Biology and Evolution, 2013.

J. Gall, Y. L. , and J. , Branching processes in L??vy processes: the exploration process, The Annals of Probability, vol.26, issue.1, pp.213-252, 1998.
DOI : 10.1214/aop/1022855417

A. Lambert and T. Stadler, Birth???death models and coalescent point processes: The shape and probability of reconstructed phylogenies, Theoretical Population Biology, vol.90, pp.113-128, 2013.
DOI : 10.1016/j.tpb.2013.10.002

URL : https://hal.archives-ouvertes.fr/hal-00936070

A. Lambert, F. Simatos, and B. Zwart, Scaling limits via excursion theory: Interplay between Crump???Mode???Jagers branching processes and processor-sharing queues, The Annals of Applied Probability, vol.23, issue.6, pp.2357-2381, 2013.
DOI : 10.1214/12-AAP904

URL : https://hal.archives-ouvertes.fr/hal-00572235

A. Lambert and P. Trapman, Splitting Trees Stopped when the First Clock Rings and Vervaat's Transformation, Journal of Applied Probability, vol.15, issue.01, pp.208-227, 2013.
DOI : 10.1214/aoap/1034625257

URL : http://arxiv.org/abs/1110.2929

G. Miermont, Ordered Additive Coalescent and Fragmentations Associated to L??vy Processes with No Positive Jumps, Electronic Journal of Probability, vol.6, issue.0, p.pp, 2001.
DOI : 10.1214/EJP.v6-87

M. Pressley-warwick, Exit properties of stochastic processes with stationary independent increments, Trans. Amer. Math. Soc, vol.178, pp.459-479, 1973.

M. Pressley-warwick, Random times and decomposition theorems, Probability (Proc. Sympos. Pure Math, pp.91-103, 1976.

[. Millar, Zero-one laws and the minimum of a Markov process, Transactions of the American Mathematical Society, vol.226, pp.365-391, 1977.
DOI : 10.1090/S0002-9947-1977-0433606-6

V. Magiorkinis, E. Sypsa, D. Magiorkinis, A. Paraskevis, R. Katsoulidou et al., Integrating Phylodynamics and Epidemiology to Estimate Transmission Diversity in Viral Epidemics, PLoS Computational Biology, vol.31, issue.2, p.1002876, 2013.
DOI : 10.1371/journal.pcbi.1002876.s009

S. Nee, R. M. May, and P. H. Harvey, The Reconstructed Evolutionary Process, Philosophical Transactions of the Royal Society B: Biological Sciences, vol.344, issue.1309, pp.305-311, 1309.
DOI : 10.1098/rstb.1994.0068

C. [. Pybus, A. Fraser, and . Rambaut, Evolutionary epidemiology: preparing for an age of genomic plenty, Philosophical Transactions of the Royal Society B: Biological Sciences, vol.368, issue.1614, pp.368-2013, 1614.
DOI : 10.1098/rstb.2012.0207

L. Popovic, Asymptotic genealogy of a critical branching process, The Annals of Applied Probability, vol.14, issue.4, pp.2120-2148, 2004.
DOI : 10.1214/105051604000000486

G. Oliver, A. Pybus, and . Rambaut, Evolutionary analysis of the dynamics of viral infectious disease, Nat Rev Genet, vol.10, issue.8, pp.540-550, 2009.

E. Pardoux and A. Wakolbinger, From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth, Electronic Communications in Probability, vol.16, issue.0, pp.720-731, 2011.
DOI : 10.1214/ECP.v16-1679

B. Rannala, Gene genealogy in a population of variable size, Heredity, vol.78, issue.4, pp.78417-423, 1309.
DOI : 10.1038/hdy.1997.65

D. A. Rasmussen, M. F. Boni, and K. Koelle, Reconciling Phylodynamics with Epidemiology: The Case of Dengue Virus in Southern Vietnam, Molecular Biology and Evolution, vol.31, issue.2, 2013.
DOI : 10.1093/molbev/mst203

. Boris-alekseevich-rogozin, On distributions of functionals related to boundary problems for processes with independent increments. Theory of Probability & Its Applications, pp.580-591, 1966.

D. A. Rasmussen, O. Ratmann, and K. Koelle, Inference for Nonlinear Epidemiological Models Using Genealogies and Time Series, PLoS Computational Biology, vol.5, issue.8, p.1002136, 2011.
DOI : 10.1371/journal.pcbi.1002136.s002

URL : http://doi.org/10.1371/journal.pcbi.1002136

D. Revuz and M. Yor, Continuous martingales and Brownian motion, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 1991.

]. S. Sca16 and . Scarpino, Evolutionary medicine IV. evolution and emergence of novel pathogens, In Encyclopedia of Evolutionary Biology, pp.77-82, 2016.

T. Stadler, D. Kühnert, S. Bonhoeffer, and A. J. Drummond, Birth-death skyline plot reveals temporal changes of epidemic spread in HIV and hepatitis C virus (HCV), Proceedings of the National Academy of Sciences, pp.228-233, 2013.
DOI : 10.1073/pnas.1207965110

[. Stadler, D. Kühnert, D. A. Rasmussen, and L. Plessis, Insights into the Early Epidemic Spread of Ebola in Sierra Leone Provided by Viral Sequence Data, PLoS Currents, 2014.
DOI : 10.1371/currents.outbreaks.02bc6d927ecee7bbd33532ec8ba6a25f

T. Stadler, On incomplete sampling under birth???death models and connections to the sampling-based coalescent, Journal of Theoretical Biology, vol.261, issue.1, pp.58-66, 2009.
DOI : 10.1016/j.jtbi.2009.07.018

T. Stadler, Inferring speciation and extinction processes from extant species data, Proceedings of the National Academy of Sciences, vol.108, issue.39, pp.16145-16146, 2011.
DOI : 10.1073/pnas.1113242108

URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3182734

M. Mark, A. R. Tanaka, F. Francis, S. A. Luciani, and . Sisson, Using approximate bayesian computation to estimate tuberculosis transmission parameters from genotype data, Genetics, vol.173, issue.3, pp.1511-1520, 2006.

. A. Elizabeth and . Thompson, Human Evolutionary Trees, 1975.

E. M. Volz, K. Koelle, and T. Bedford, Viral Phylodynamics, PLoS Computational Biology, vol.151, issue.3, p.1002947, 2013.
DOI : 10.1371/journal.pcbi.1002947.s002

URL : http://doi.org/10.1371/journal.pcbi.1002947

E. M. Volz, S. L. Kosakovsky-pond, M. J. Ward, A. J. Leigh-brown, and S. D. Frost, Phylodynamics of Infectious Disease Epidemics, Genetics, vol.183, issue.4, pp.1421-1430, 2009.
DOI : 10.1534/genetics.109.106021

URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2787429

N. D. Wolfe, C. P. Dunavan, and J. Diamond, Origins of major human infectious diseases, Nature, vol.10, issue.7142, pp.279-83, 2007.
DOI : 10.1038/nature05775

D. Williams, Path decomposition and continuity of local time for onedimensional diffusions. I, Proc. London Math. Soc. (3), pp.738-768, 1974.

I. Ahmed and . Zayed, Handbook of function and generalized function transformations Mathematical Sciences Reference Series, 1996.