For a positive integer $d$, every finite subcubic graph with edge expansion at least $1/(8d)$ satisfies $|V|\le(24d+1)(\operatorname{tw}(G)+1)$.

If every component after deleting $X$ has size at most $|V|/2$, summing the expansion inequalities over components gives $|V\setminus X|\le24d|X|$. Thus a smaller deletion leaves a component containing more than half the graph. These large connected sets form a bramble. A bag of every tree decomposition hits the bramble, giving the bound.