  Tree-like tableaux are combinatorial objects that appear in a combinatorial
understanding of the PASEP model from statistical mechanics. In this
understanding, the corners of the Southeast border correspond to the locations
where a particle may jump to the right. Such corners may be of two types:
either empty or occupied. Our main result is the following: on average there is
one occupied corner per tree-like tableau. We give two proofs of this, a short
one which gives us a polynomial version of the result, and another one using a
bijection between tree-like tableaux and permutations which gives us an
additional information. Moreover, we obtain the same result for symmetric
tree-like tableaux and we refine our main result to an equivalence class.
Finally we present a conjecture enumerating corners, and we explain its
consequences for the PASEP.
