  This is the second of a series of four papers in which we prove the following
relaxation of the Loebl-Komlos--Sos Conjecture: For every $\alpha>0$ there
exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$
with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$
contains each tree $T$ of order $k$ as a subgraph.
  In the first paper of the series, we gave a decomposition of the graph $G$
into several parts of different characteristics; this decomposition might be
viewed as an analogue of a regular partition for sparse graphs. In the present
paper, we find a combinatorial structure inside this decomposition. In the last
two papers, we refine the structure and use it for embedding the tree $T$.
