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The Eulerian polynomials $A_n(x)$ give the distribution of descents overpermutations. It is also known that the distribution of descents overstack-sortable permutations (i.e. permutations sortable by a certain algorithmwhose internal storage is limited to a single stack data structure) is given bythe Narayana numbers $\frac{1}{n}{n \choose k}{n \choose k+1}$. On the otherhand, as a corollary of a much more general result, the distribution of thestatistic ``maximum number of non-overlapping descents'', MND, over allpermutations is given by $\sum_{n,k \geq 0}D_{n,k}x^k\frac{t^n}{n!}=\frac{e^t}{1-x(1+(t-1)e^t)}$.  In this paper, we show that the distribution of MND over stack-sortablepermutations is given by $\frac{1}{n+1}{n+1\choose 2k+1}{n+k \choose k}$. Wegive two proofs of the result via bijections with rooted plane (binary) treesallowing us to control MND. Moreover, we show combinatorially that MND isequidistributed with the statistic MNA, the maximum number of non-overlappingascents, over stack-sortable permutations. The last fact is obtained byestablishing an involution on stack-sortable permutations that givesequidistribution of 8 statistics.

Non-overlapping descents and ascents in stack-sortable permutations