Revisiting John Lamperti's maximal branching process

Lamperti's maximal branching process is revisited, with emphasis on the description of shape invariant measures in both recurrent and transient regimes. A truncated version this chain exhibited, preserving monotonicity original Lamperti supported by integers. The Brown theory hitting times applies to latter finite state-space, including sharp strong time stationarity. Additional information these problems are drawn from quasi-stationary point view.