  The purpose of the paper is to present quantitative estimates for the
principal eigenvalue of discrete p-Laplacian on the set of rooted trees.
Alternatively, it is studying the optimal constant of a class of weighted Hardy
inequality. Three kinds of variational formulas in different formulation for
the mixed principal eigenvalue of p-Laplacian on the set of trees with unique
root as Dirichlet boundary are presented. As their applications, we obtain a
basic estimate of the eigenvalue on trees.
