  We prove an ergodic theorem for Markov chains indexed by the
Ulam-Harris-Neveu tree over large subsets with arbitrary shape under two
assumptions: with high probability, two vertices in the large subset are far
from each other and have their common ancestor close to the root. The
assumption on the common ancestor can be replaced by some regularity assumption
on the Markov transition kernel. We verify that those assumptions are satisfied
for some usual trees. Finally, with Markov-Chain Monte-Carlo considerations in
mind, we prove when the underlying Markov chain is stationary and reversible
that the Markov chain, that is the line graph, yields minimal variance for the
empirical average estimator among trees with a given number of nodes.
