  We consider the bar complex of a monomial non-unital associative algebra $A=k
\langle X \rangle / (w_1,...,w_t)$. It splits as a direct sum of complexes
$B_w$, defined for any fixed monomial $w=x_1...x_n \in A$.
  We give a simple argument, showing that the homology of this subcomplex is at
most one-dimensional, and describe the place where the nontrivial homology
appears. It has a very simple expression in terms of the length of the
generalized Dyck path associated to a given monomial in $w \in A$.
  The operadic analogue of the question about dichotomy in homology is
considered. It is shown that dichotomy holds in case when monomial
tree-relations form an order. Examples are given showing that in general
dichotomy and homological purity does not hold. For quadratic operads, the
combinatorial tool for calculating homology in terms of relation graphs is
developed. Example of using these methods to compute homology in truncated
binary operads is given.
