  Many proposed scenarios for black hole (BH) mergers involve a tertiary
companion that induces von Zeipel-Lidov-Kozai (ZLK) eccentricity cycles in the
inner binary. An attractive feature of such mechanisms is the enhanced merger
probability when the octupole-order effects, also known as the eccentric Kozai
mechanism, are important. This can be the case when the tertiary is of
comparable mass to the binary components. Since the octupole strength [$\propto
(1-q)/(1+q)$] increases with decreasing binary mass ratio $q$, such ZLK-induced
mergers favor binaries with smaller mass ratios. We use a combination of
numerical and analytical approaches to fully characterize the octupole-enhanced
binary BH mergers and provide analytical criteria for efficiently calculating
the strength of this enhancement. We show that for hierarchical triples with
semi-major axis ratio $a/a_{\rm out}\gtrsim 0.01$-$0.02$, the binary merger
fraction can increase by a large factor (up to $\sim 20$) as $q$ decreases from
unity to $0.2$. The resulting mass ratio distribution for merging binary BHs
produced in this scenario is in tension with the observed distribution obtained
by the LIGO/VIRGO collaboration, although significant uncertainties remain
about the initial distribution of binary BH masses and mass ratios.
