Trees in Wavelet analysis on Vilenkin groups

We consider a class of $(1,M)$-elementary step functions on the $p$-adic Vilenkin group. We prove that $(1,M)$-elementary step function generates a MRA on $p$-adic Vilenkin group iff it is generated by a rooted tree on the set of vertices $\{0,1,\dots p-1\}$ with 0 as a root. Bibliography: 14 titles.