We discuss a new notion of pattern avoidance motivated by the operad theory:
pattern avoidance in planar labelled trees. It is a generalisation of various
types of consecutive pattern avoidance studied before: consecutive patterns in
words, permutations, coloured permutations etc. The notion of Wilf equivalence
for patterns in permutations admits a straightforward generalisation for (sets
of) tree patterns; we describe classes for trees with small numbers of leaves,
and give several bijections between trees avoiding pattern sets from the same
class.

We show how to use Groebner bases for operads to prove various freeness
theorems: freeness of certain operads as nonsymmetric operads, freeness of an
operad Q as a P-module for an inclusion P into Q, freeness of a suboperad. This
gives new proofs of many known results of this type and helps to prove some new
results.

We define a new monoidal category on collections (shuffle composition).
Monoids in this category (shuffle operads) turn out to bring a new insight in
the theory of symmetric operads. For this category, we develop the machinery of
Gr\"obner bases for operads, and present operadic versions of Bergman's Diamond
Lemma and Buchberger's algorithm. This machinery can be applied to study
symmetric operads. In particular, we obtain an effective algorithmic version of
Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads.