  In this work we introduce and study tree-like tableaux, which are certain
fillings of Ferrers diagrams in simple bijection with permutation tableaux and
alternative tableaux. We exhibit an elementary insertion procedure on our
tableaux which gives a clear proof that tree-like tableaux of size n are
counted by n!, and which moreover respects most of the well-known statistics
studied originally on alternative and permutation tableaux. Our insertion
procedure allows to define in particular two simple new bijections between
tree-like tableaux and permutations: the first one is conceived specifically to
respect the generalized pattern 2-31, while the second one respects the
underlying tree of a tree-like tableau.
