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We consider a population with non-overlapping generations, whose size goes toinfinity. It is described by a discrete genealogy which may be timenon-homogeneous and we pay special attention to branching trees in varyingenvironments. A Markov chain models the dynamic of the trait of each individualalong this genealogy and may also be time non-homogeneous. Such models aremotivated by transmission processes in the cell division,reproduction-dispersion dynamics or sampling problems in evolution. We want todetermine the evolution of the distribution of the traits among the population,namely the asymptotic behavior of the proportion of individuals with a giventrait. We prove some quenched laws of large numbers which rely on theergodicity of an auxiliary process, in the same vein as \cite{guy,delmar}.Applications to time inhomogeneous Markov chains lead us to derive a backward(with respect to the environment) law of large numbers and a law of largenumbers on the whole population until generation $n$. A central limit is alsoestablished in the transient case.

Weak law of large numbers for some Markov chains along non homogeneous genealogies