A characteristic property of spheres in regular trees (Q5936060)

Let \(x\) be a vertex and \(T\) a finite subtree of a regular tree \(\mathcal T_k\). In the paper it is proved that if for every harmonic function \(f\) on \(\mathcal T_k\), \(f(x)\) equals the mean value of \(f\) on the leaves of \(T\), then these leaves form a sphere in \(\mathcal T_k\) with center \(x\).