It is well known that an arbitrary graphical model of statistical inference defined 
on a tree, i.e. on a graph without loops, is solved exactly and efficiently by an 
algorithm of the Belief Propagation (BP) type. Extending recent results of 
Kolmogorov, Wainwright '05 and Bayati, Shah, Sharma '06, we discuss here 
two cases of the opposite extreme, where  BP algorithm finds optimal Maximum 
Likelihood solution for special models on graphs with loops. Defining BP solution 
at a finite temperature as the global minimum of the Bethe free energy, 
we show that in the limit of zero temperature the two models are solved exactly by BP.