  We study the Lane-Emden problem \[\begin{cases}
  -\Delta u_p =|u_p|^{p-1}u_p&\text{in}\quad \Omega,
  u_p=0 &\text{on}\quad\partial\Omega,
  \end{cases}\] where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and
$p>1$ is sufficiently large. We obtain sharp estimates and non-degeneracy of
low energy nodal solutions $u_p$ (i.e. nodal solutions satisfying
$\lim_{p\to+\infty}p\int_{\Omega}|u_p|^{p+1}dx=16\pi e$). As applications, we
prove that the comparable condition
$p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$ holds automatically for least
energy nodal solutions, which confirms a conjecture raised by
(Grossi-Grumiau-Pacella, Ann.I.H. Poincare-AN, 30 (2013), 121-140) and
(Grossi-Grumiau-Pacella, J.Math.Pures Appl. 101 (2014), 735-754).
