  We study the possibility of generating non-zero reactor mixing angle
$\theta_{13}$ by perturbing the $\mu-\tau$ symmetric neutrino mass matrix. The
leading order $\mu-\tau$ symmetric neutrino mass matrix originates from type I
seesaw mechanism whereas the perturbations to $\mu-\tau$ symmetry originate
from type II seesaw term. We consider four different realizations of $\mu-\tau$
symmetry: Bimaximal Mixing(BM), Tri-bimaximal Mixing (TBM), Hexagonal Mixing
(HM) and Golden Ratio Mixing (GRM) all giving rise to $\theta_{13} = 0,
\theta_{23} = \frac{\pi}{4}$ but different non-zero values of solar mixing
angle $\theta_{12}$. We assume a minimal $\mu-\tau$ symmetry breaking type II
seesaw mass matrix as a perturbation and calculate the neutrino oscillation
parameters as a function of type II seesaw strength. We then consider the
origin of non-trivial leptonic CP phase in the charged lepton sector and
calculate the lepton asymmetry arising from the lightest right handed neutrino
decay by incorporating the presence of both type I and type II seesaw. We
constrain the type II seesaw strength as well as leptonic CP phase (and hence
the charged lepton sector) by comparing our results with experimental neutrino
oscillation parameters as well as Planck bound on baryon to photon ratio.
Finally, we extend our analysis on lepton flavour violating decays like $\mu
\to e \gamma$ and $\mu \to eee$ due to exchange of TeV scale Higgs triplet
scalar within the low scale type II seesaw framework. The branching ratios for
these lepton flavour processes are examined with the small type II perturbation
term $\omega$ and the estimated values are very close to the experimental bound
coming from current search experiments.
