Growing conditioned BGW trees with log-concave offspring distributions

We show that given a log-concave offspring distribution, the corresponding sequence of Bienaymé-Galton-Watson trees conditioned to have $n\ge 1$ vertices admits a realization as a Markov process $(T_n{)}_{n\ge 1}$ which adds a new “right-leaning” leaf at each step. This applies for instance to offspring distributions which are Poisson, binomial, geometric, or any convolution of those. By a negative result of Janson, the log-concavity condition is optimal in the restricted case of offspring distributions supported in $\{0,1,2\}$. We then prove a generalization to the case of an offspring distribution supported on an arithmetic progression, if we assume log-concavity along that progression. As an application, we deduce the existence of increasing couplings in an inhomogeneous model of random subtrees of the Ulam–Harris tree. This is equivalent to the statement that, in a corresponding inhomogeneous Bernouilli percolation model on a regular tree, the root cluster is stochastically increasing in its size. These results generalize a construction of Luczak and Winkler which applies to uniformly sampled subtrees with $n$ vertices of the infinite complete $d$-ary trees. Our proofs are elementary and we tried to make them as self-contained as possible.