A Markovian growth dynamics on rooted binary trees evolving according to the Gompertz curve

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $\beta>0$. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate $\beta(n-k)/n$, where $k$ is the distance from the node to the root. Denote by $Z_n(t)$ the number of nodes with no descendants at time $t$ and let $T_n = \beta^{-1} n \ln(n /\ln 4) + (\ln 2)/(2 \beta)$. We prove that $2^{-n} Z_n(T_n + n \tau)$, $\tau\in\bb R$, converges to the Gompertz curve $\exp (- (\ln 2) e^{-\beta \tau})$. We also prove a central limit theorem for the martingale associated to $Z_n(t)$.