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 (1/2 + 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 re finement of this structure. In the fourth paper, the re fined structure will be used for embedding the tree T.