**A Necessary and Sufficient Condition for the Tail-Triviality of a
Recursive Tree Process**

**Abstract :**
Given a *recursive distributional equation* (RDE) and a solution $\mu$
of it, we consider the tree indexed invariant process called the
*recursive tree process* (RTP) with marginal $\mu$.
We introduce a new type of bivariate uniqueness property which is
different from the one defined by Aldous and Bandyopadhyay (2005),
and we prove that this property is equivalent to tail-triviality for the
RTP, thus obtaining a necessary and sufficient condition to determine
tail-triviality for a RTP in general. As an application we consider
Aldous' (2000) construction of the frozen percolation process on a infinite
regular tree and show that the associated RTP has a trivial tail.