  This is the third 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. In the second paper, we found
a combinatorial structure inside the decomposition. In this paper, we will give
a refinement of this structure. In the forthcoming fourth paper, the refined
structure will be used for embedding the tree $T$.
