  In this note we give several counterexamples. One shows that small energy
majorization on bi-tree fails. The second counterexample shows that partial
energy estimate 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 partial energy estimate called surrogate maximum
principle: $\int_{T^2} V^\nu_\varepsilon \, d\nu \le C_\tau
\varepsilon^{1-\tau} {\mathcal E}[\nu]^{\tau} |\nu|^{1-\tau}$ is valid on
bi-tree with any $\tau>0$. We show that unlike the estimate on a simple tree,
one cannot make $\tau=0$ on bi-tree.
  On tri-tree we know that the previous estimate (the surrogate maximum
principle) is valid with $\tau=2/3$. We do not know any such estimate with any
$\tau<1$ on four-tree.
  The third counterexample disproves the estimate $\int_{T^2} V^\nu_x \, d\nu
\le F(x)$ for any function $F$ whatsoever for some probabilistic $\nu$ on
bi-tree $T^2$. On a simple tree $F(x)=x$ would always suffice to make this
inequality to hold.
