  In this paper, we have used the state finder hierarchy for Barrow holographic
dark energy (BHDE) model in the system of the FLRW Universe. We apply two DE
diagnostic tools to determine $\Lambda BHDE$ model to have various estimations
of $\triangle$. The first diagnostic tool is the state finder hierarchy in
which we have studied $S^{(1)}_{3}$ $S^{(1)}_{4}$, $S^{(2)}_{3}$ $S^{(2)}_{4}$
and second is the composite null diagnostic (CND) in which the trajectories (
$S^{(1)}_{3} - \epsilon$), ($S^{(1)}_{4}-\epsilon$),
($S^{(2)}_{3}-\epsilon$)($S^{(2)}_{4}-\epsilon$) are discussed, where
$\epsilon$ is fractional growth parameter. The infrared cut-off here is taken
from the Hubble horizon. Finally, together with the growth rate of matter
perturbation, we plot the state finder hierarchy that determines a composite
null diagnosis that can differentiate emerging dark energy from $\Lambda CDM$.
In addition, the combination of the hierarchy of the state finder and the
fractional growth parameter has been shown to be a useful tool for diagnosing
BHDE, particularly for breaking the degeneracy with different parameter values
of the model.
