Suppose there is a population of trees, and in each period, each tree faces a probability $p$ of getting cut down. If a tree is cut down, its height is reset to zero (it doesn't die, this is merely a setback for it); if a tree is not cut down, then its height increases by $1$ unit. Namely, if $h_{t}$ is the height of some tree, then $$ h_{t+1} = \begin{cases} h_{t}+1 & \textrm{with probability } 1-p; \\ 0 & \textrm{with probability } p. \end{cases} $$ What I am looking for is the stationary probability distribution $f(h)$ (for the forest as a whole); I initially thought this would be some trivial extension of the geometric/exponential distribution (ideally I would find a result for continuous time), maybe using convolution? but now I am less sure.
I found this question, but it doesn't seem to be quite what I'm looking for. I've also started reading William Feller's An Introduction to Probability Theory and Its Applications, so perhaps that will help.
Any suggestions/leads would be appreciated; part of the difficulty has just been figuring out the relevant terms to search for…