  The celebrated dependent random choice lemma states that in a bipartite graph
an average vertex (weighted by its degree) has the property that almost all
small subsets $S$ in its neighborhood has common neighborhood almost as large
as in the random graph of the same edge-density. Two well-known applications of
the lemma are as follows. The first is a theorem of F\"uredi and of Alon,
Krivelevich, and Sudakov showing that the maximum number of edges in an
$n$-vertex graph not containing a fixed bipartite graph with maximum degree at
most $r$ on one side is $O(n^{2-1/r})$. This was recently extended by Grzesik,
Janzer and Nagy to the family of so-called $(r,t)$-blowups of a tree. A second
application is a theorem of Conlon, Fox, and Sudakov, confirming a special case
of a conjecture of Erd\H{o}s and Simonovits and of Sidorenko, showing that if
$H$ is a bipartite graph that contains a vertex complete to the other part and
$G$ is a graph then the probability that the uniform random mapping from $V(H)$
to $V(G)$ is a homomorphismis at least
$\left[\frac{2|E(G)|}{|V(G)|^2}\right]^{|E(H)|}$.
  In this note, we introduce a nested variant of the dependent random choice
lemma, which might be of independent interest. We then apply it to obtain a
common extension of the theorem of Conlon, Fox, and Sudakov and the theorem of
Grzesik, Janzer, and Nagy, regarding Tur\'an and Sidorenko properties of
so-called tree-degenerate graphs.
