  In this paper, we study tree--like tableaux, combinatorial objects which
exhibit a natural tree structure and are connected to the partially asymmetric
simple exclusion process (PASEP). There was a conjecture made on the total
number of corners in tree--like tableaux and the total number of corners in
symmetric tree--like tableaux. In this paper, we prove the first conjecture
leaving the proof of the second conjecture to the full version of this paper.
Our proofs are based on the bijection with permutation tableaux or type--B
permutation tableaux and consequently, we also prove results for these
tableaux.
