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Loebl, Koml\'os and S\'os conjectured that every $n$-vertex graph $G$ with atleast $n/2$ vertices of degree at least $k$ contains each tree $T$ of order$k+1$ as a subgraph. We give a sketch of a proof of the approximate version ofthis conjecture for large values of $k$.  For our proof, we use a structural decomposition which can be seen as ananalogue of Szemer\'edi's regularity lemma for possibly very sparse graphs.With this tool, each graph can be decomposed into four parts: a set of verticesof huge degree, regular pairs (in the sense of the regularity lemma), and twoother objects each exhibiting certain expansion properties. We then exploit theproperties of each of the parts of $G$ to embed a given tree $T$.  The purpose of this note is to highlight the key steps of our proof. Detailscan be found in [arXiv:1211.3050].

The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs