Detailed balance is the condition that, on average, for each pair of states z and z , the number of transitions between the two states is the same in either direction: where is the transition performed by one step of the sweep: It is important to observe that, because the resampling step only changes the tree structure below the ith node, the ith node in the new tree remains the same node.
that is, after making an update from z to z , v(z, i)  , and, mathematically: the condition in Equation 8is symmetric in z and z , and we define the predicate match(z, z , i) to be equivalent to this condition.