  Fix $n\in\mathbb{N}$. Let $\mathbf{T}_n$ be the set of rooted trees $(T,o)$
whose vertices are labeled by elements of $\{1,...,n\}$. Let $\nu$ be a
strongly connected multi-type Galton-Watson measure. We give necessary and
sufficient conditions for the existence of a measure $\mu$ that is reversible
for simple random walk on $\mathbf{T}_n$ and has the property that given the
labels of the root and its neighbors, the descendant subtrees rooted at the
neighbors of the root are independent multi-type Galton-Watson trees with
conditional offspring distributions that are the same as the conditional
offspring distributions of $\nu$ when the types are $\nu$ are ordered pairs of
elements of $[n]$. If the types of $\nu$ are given by the labels of vertices,
then we give an explicit description of such $\mu$.
