A completely hyperexpansive completion problem for weighted shifts on directed trees with one branching vertex

Let ? = {?k}nk =0 be given a finite sequence of positive real numbers. The completely hyperexpansive completion problem seeks equivalence conditions for the existence of a completely hyperexpansive weighted shift W?? such that ? ? ??. Let T?,? be a directed tree consisting of one branching vertex, ? branches and a trunk of length ?, and let T?,?,p be a subtree of T?,? whose members consist of the p-generation family from branching vertex. Suppose S? is the weighted shift acting on the tree T?,?. This object S? on the tree T?,? has been applied to the several topics. Recently, Exner-Jung-Stochel-Yun studied the subnormal completion problem for weighted shifts on T?,? in 2018. In this paper we discuss the completely hyperexpansive completion problem for weighted shifts on T?,? as a counterpart of the subnormal completion problem for S?.