Differences between the potential theories on a tree and on a bi-tree

In this note we give several counterexamples. One shows that small energy majorization on bi-tree fails. The second counterexample shows that energy estimate $\int_T \bV^\nu_\eps \, d\nu \le C \eps |\nu|$ always valid on a usual tree by a trivial reason (and with constant $C=1$) cannot be valid in general on bi-tree with any $C$ whatsoever. On the other hand, a weaker estimate $\int_{T^2} \bV^\nu_\eps \, d\nu \le C_\tau \eps^{1-\tau} \cE[\nu]^{\tau} |\nu|^{1-\tau}$ is valid on bi-tree with any $\tau>0$. It is proved in \cite{MPVZ} and is called improved surrogate maximum principle for potentials on bi-tree. The estimate $\int_{T^3} \bV^\nu_\eps \, d\nu \le C_\tau \eps^{1-\tau} \cE[\nu]^{\tau} |\nu|^{1-\tau}$ with $\tau=2/3$ holds on tri-tree. We do not know {\it any} such estimate with {\it} any $\tau<1$ on four-tree. The third counterexample disproves the estimate $\int_{T^2} \bV^\nu_x \, d\nu \le F(x)$ for any $F$ whatsoever for some probabilistic $\nu$ on bi-tree $T^2$. On a simple tree $F(x)=x$ would suffice to make this inequality to hold.