  We present analysis of two- and three-point correlation functions of
Ly$\alpha$ forest (at $2\le z\le 2.5$) using X-Shooter spectra of three
background quasar triplets probing transverse separations of 0.5-1.6 pMpc. We
present statistics based on transmitted flux and clouds identified using Voigt
profile fitting. We show that the observed two-, three-point correlation
functions and reduced three-point correlation (i.e Q) are well reproduced by
our simulations. We assign probabilities for realising all the observed
correlation properties simultaneously using our simulations. Our simulations
suggest an increase in correlation amplitudes and Q with increasing $N_{\rm
HI}$. We roughly see this trend in the observations too. We identify a
concurrent gap of 17$\mathring{A}$ (i.e 14.2 $h^{-1}$cMpc, one of the longest
reported) wide in one of the triplets. Such gap is realised only in 14.2% of
our simulated sightlines and most of the time belongs to a void in the matter
distribution. In the second triplet, we detect DLAs along all three sightlines
(with spatial separations 0.64 to 1.6 pMpc) within a narrow redshift interval
(i.e $\Delta z = 0.088$). Detection of a foreground quasar ($\sim$ 1 pMpc from
the triplet sightlines) and excess partial Lyman Limit systems around these
DLAs suggest that we may be probing a large over-dense region. We also report
positive CIV- CIV correlations up to $\sim 500$ $km s^{-1}$ only in the
longitudinal direction. Additionally, we conclude a positive CIV-Ly$\alpha$
correlations for higher $N_{\rm HI}$ thresholds up to a scale of $\sim 1000$
$km s^{-1}$ both in transverse and longitudinal directions.
