The universal prevalence of online social networks (OSNs) is unquestionable today; the interesting posts/contents are forwarded repeatedly by the users of the OSN. Such propagation leads to the re-forwarding of the post to a subset of previous recipients, which increases as the post gets viral (reaches a large number of the users). Consequently, the effective forwards (after deleting the re-forwards) reduce, eventually leading to the saturation of the total number of copies. We study such a process using a new variant of the branching process and named it as 'Saturated total-population-dependent branching process'.We analyse the new variant using the stochastic approximation technique and derive an ODE, dependent on the expected effective forwards. Finally, we obtain deterministic trajectories which approximate the total and unread copies of the post 'asymptotically and almost surely' over any finite time window; this trajectory is entirely described by four parameters related to the network characteristics. Further, we provide theoretical expressions for the peak unread copies, maximum outreach and the life span of the post. We again observe well-known exponential growth, however, with time-varying rates. At last, we validate our theory through detailed simulations on the SNAP Twitter dataset, which illustrate a good fit. Recently, we analysed many new variants of BPs, using a new approach based on stochastic approximation techniques, e.g., attack and acquisition BP (competing viral markets [5]), and proportion dependent BP (fake news on OSNs [7]).Further modifying the said approach to address the required finite horizon analysis, we are able to analyse saturation resulting from total population dependency. Key contributions: The contribution of this paper is two fold: first, we formally analyse the STP-BP. We derive an appropriate ordinary differential equation (ODE) and its solution which (time asymptotically) almost surely approx-