  Treedepth, a more restrictive graph width parameter than treewidth and
pathwidth, plays a major role in the theory of sparse graph classes. We show
that there exists a constant $C$ such that for every positive integers $a,b$
and a graph $G$, if the treedepth of $G$ is at least $Cab$, then the treewidth
of $G$ is at least $a$ or $G$ contains a subcubic (i.e., of maximum degree at
most $3$) tree of treedepth at least $b$ as a subgraph.
  As a direct corollary, we obtain that every graph of treedepth $\Omega(k^3)$
is either of treewidth at least $k$, contains a subdivision of full binary tree
of depth $k$, or contains a path of length $2^k$. This improves the bound of
$\Omega(k^5 \log^2 k)$ of Kawarabayashi and Rossman [SODA 2018].
  We also show an application of our techniques for approximation algorithms of
treedepth: given a graph $G$ of treedepth $k$ and treewidth $t$, one can in
polynomial time compute a treedepth decomposition of $G$ of width
$\mathcal{O}(kt \log^{3/2} t)$. This improves upon a bound of $\mathcal{O}(kt^2
\log t)$ stemming from a tradeoff between known results.
  The main technical ingredient in our result is a proof that every tree of
treedepth $d$ contains a subcubic subtree of treedepth at least $d \cdot \log_3
((1+\sqrt{5})/2)$.
