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We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with$\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binarytree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcalT_k$ is the $k$th generation of $\mathcal T$. In addition, let$(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in\mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markovchain $\mathcal X^n$ along the random walk. We present a law of large numbersconcerning the empirical measure process $\widetilde{\mathcal Z}^n =(\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n =\sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$.Precisely, we show that if $\widetilde{\mathcal R}^n \to \mathcal R$ for someFeller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initialcondition, then $\widetilde{\mathcal Z}^n \to \mathcal Z$ with $Z_t =\delta_{\mathcal L(R_t)}$.

Some limit results for Markov chains indexed by trees