Let α = {α k } n k=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 ˆ α such that α ⊆ ˆ α. Let T η,κ be a directed tree consisting of one branching vertex, η branches and a trunk of length κ, and let T η,κ,p be a subtree of T η,κ whose members consist of the p-generation family from branching vertex. Suppose S λ is the weighted shift acting on the tree T η,κ. This object S λ on the tree T η,κ has been applied to the several topics. Recently, Exner-Jung-Stochel-Yun studied the subnormal completion problem for weighted shifts on T η,κ in 2018. In this paper we discuss the completely hyperexpansive completion problem for weighted shifts on T η,κ as a counterpart of the subnormal completion problem for Sλ .