Polynomial functors, trees, and opetopes

Polynomial functors are certain functors built from disjoint union,
product, and exponentiation.  They are a categorification of the notion of
polynomial function: you can add and multiply them or substitute them into
each other; you can also differentiate them, and there is a Leibniz rule
and a chain rule.  All these manipulations can be done just in terms of
coefficients and exponents.  After this introductory material, I will
survey some ways in which polynomial functors are intimately linked with
trees and operads.  Trees arise from free monads on polynomial
endofunctors.  For such, the disjoint union of all coeffcients is a
collection of (decorated) trees, and they correspond to the set of
operations of an (coloured) operad, while the exponents correspond to their
arities.  There is also a canonical way of associating a polynomial functor
to a tree, and a characterisation of the polynomial functors that arise in
this way; hence the very notion of tree is just a special case of
polynomial functor.  To finish with some recent research (joint with Joyal,
Batanin, and Mascari), I will explain how polynomial functors provide an
elegant construction of 'higher trees', the so-called opetopes, which are
the basis for the Baez-Dolan approach to higher category theory.