  Loebl, Koml\'os and S\'os conjectured that every $n$-vertex graph $G$ with at
least $n/2$ 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 of
this conjecture for large values of $k$.
  For our proof, we use a structural decomposition which can be seen as an
analogue 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 vertices
of huge degree, regular pairs (in the sense of the regularity lemma), and two
other objects each exhibiting certain expansion properties. We then exploit the
properties 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. Details
can be found in [arXiv:1211.3050].
