  Assume that \Gamma_{v_0} is a tree with vertex set Vert(\Gamma_{v_0})={v_0,
v_1,..., v_n}, and with an integral framing (weight) attached to each vertex
except v_0. Assume furthermore that the intersection matrix of
G=\Gamma_{v_0}-{v_0} is negative definite. We define a filtration on the chain
complex computing the lattice homology of G and show how to use this
information in computing lattice homology groups of a negative definite graph
we get by attaching some framing to v_0. As a simple application we produce
families of graphs which have arbitrarily many bad vertices for which the
lattice homology groups are shown to be isomorphic to the corresponding
Heegaard Floer homology groups.
