In classical Morse theory, for any given manifold there is always a unique optimal Morse vector (=the vector counting the number of critical points of index 0,1,..., up to the dimension). It turns out that in Forman's discrete version of Morse theory, this is no longer the case. I will sketch how to construct a contractible 3-complex on which the 'best' discrete Morse vectors are (1,0,1,1) and (1,1,1,0), because actually (1,0,0,0) is out of reach. On the way to this result, we need to answer to the following question: How does one generalize to all dimensions the well-known statement "every (finite) tree has at least two leaves?"

This is joint work with Karim A. Adiprasito and Frank H. Lutz.