The unique continuation property for a nonlinear equation on trees

In this paper we study the game $p-$Laplacian on a tree, that is, $$u(x)=\frac{\alpha}2\left\{\max_{y\in \S(x)}u(y) + \min_{y\in \S(x)}u(y)\right\} + \frac{\beta}{m}\sum_{y\in \S(x)} u(y),$$ here $x$ is a vertex of the tree and $S(x)$ is the set of successors of $x$. We study the family of the subsets of the tree that enjoy the unique continuation property, that is, subsets $U$ such that $u\mid_U=0$ implies $u \equiv 0$.