Hello,

I'm relatively new to this forum so apologies if I have tagged my question incorrectly.

I have been in contact with a wildlife biologist recently concerning counting elephant populations and I wonder if people could comment on the following approach.

de Bruijn, Knuth and Rice showed that the expected height of a general Catalan tree is $\sqrt{\pi} \sqrt{n}$. In terms of Dyck words this translates to saying that the expected maximum excess of Xs over Ys in a Dyck word of length $2n$ is $\sqrt{\pi} \sqrt{n}$.

Now if we think of X as "an elephant arrives at the watering hole" and Y as the elephant leaving it then should not the largest number seen correspond to the above expression? Thus we could get an estimate of the population. And that is my question, how good an approximation to the population $n$ will this be?

Note that it is not a concern that an elephant revisits the site and is doubly counted, because this will not affect the maximum seen. However, one flaw in the model is that it assumes that elephants arrive independently. Certainly this will not be the case with young calves but there may be cliques who are fellow travellers also. Also, perhaps, the model assumes that the likelihood of an elephant being present at time $t$ is equally likely for all $t$. This may not be realistic either.

Apologies also to those who find all of the above just elephants :-).

Thanks, Patrick healy