  We investigate the doubling inequality and nodal sets for the solutions of
bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions
and their gradient is obtained based on the recent development of nodal sets
for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper
bound for the nodal sets of solutions. We show two types of doubling
inequalities for the solutions of bi-Laplace equations. As a consequence, the
rate of vanishing is given for the solutions.
