  This paper is concerned with certain generalizations of meagreness and their
combinatorial equivalents. The simplest example, and the one which motivated
further study in this area, comes about by considering the following
definition: a set X subseteq R is said to be Q-nowhere dense if and only if for
every rational q there exists and integer k such that the interval whose
endpoints are q and q+1/k is disjoint from X. A set which is the union of
countably many Q-nowhere dense sets will be called Q-very meagre.
  Steprans considered the least number of Q-meagre sets required to cover the
real line and denoted by d_1. He showed that there is a continuous function H
--- first constructed by Lebesgue --- such that the least number of smooth
functions into which H can be decomposed is equal to d_1. This paper will
further study d_1 and some of its generalizations. As well, an equivalence will
be established between Q-meagreness and certain combinatorial properties of
trees. This will lead to new cardinal invariants and various independence
results about these will then be established.
