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Denote by ${𝒯}_n$, $n\ge 2$, the regular tree whose vertices have valence $n+1$, $\partial {𝒯}_n$ its boundary. Yu. A. Neretin has proposed a group $N_n$ of transformations of $\partial {𝒯}_n$, thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that $N_n$ is generated by two groups: the group $\mathrm{Aut}\left({𝒯}_n\right)$ of tree automorphisms, and a Higman-Thompson group $G_n$. We prove the simplicity of $N_n$ and of a family of its subgroups.