  We explore the relationship between polynomial functors and (rooted) trees.
In the first part we use polynomial functors to derive a new convenient
formalism for trees, and obtain a natural and conceptual construction of the
category $\Omega$ of Moerdijk and Weiss; its main properties are described in
terms of some factorisation systems. Although the constructions are motivated
and explained in terms of polynomial functors, they all amount to elementary
manipulations with finite sets. In the second part we describe polynomial
endofunctors and monads as structures built from trees, characterising the
images of several nerve functors from polynomial endofunctors and monads into
presheaves on categories of trees. Polynomial endofunctors and monads over a
base are characterised by a sheaf condition on categories of decorated trees.
In the absolute case, one further condition is needed, a certain projectivity
condition, which serves also to characterise polynomial endofunctors and monads
among (coloured) collections and operads.
