Abstract: Lie commutator and Jordan commutator are defined by [a,b]=ab-ba and {a,b}=ab+ba. There are two kind of generalizations of commutators. The first one is q-commutator ab+q ba, where q is not necessary 1 or -1. The second one is N-ary commutator: skew-symmetric sum of N! compositions. We explain why Lie and Jordan commutators are more popular than q-commutators. S. Lie noticed that commutator of vector fields is a well defined operation. As it turned out Lie commutator is not unique operation on a space of vector fields. We show that vector fields on n-dimensional manifolds may have new N-ary commutators for N=N(n)>2. We explain how to calculate compositions of vector fields and how to calculate powers of differential operators in terms of rooted trees. We develop super-tree calculus and discuss its connection with permutation statistics. We construct differential algebraic structures with identities that generalise Lie, Jordan and Jacobi identities.
