  This article investigates combinatorial properties of non-ambiguous trees.
These objects we define may be seen either as binary trees drawn on a grid with
some constraints, or as a subset of the tree-like tableaux previously defined
by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees
satisfying some additional constraints allows us to give elegant combinatorial
proofs of identities due to Carlitz, and to Ehrenborg and Steingr\'imsson. We
also provide a hook formula to count the number of non-ambiguous trees with a
given underlying tree. Finally, we use non-ambiguous trees to describe a very
natural bijection between parallelogram polyominoes and binary trees.
