In this paper, we firstly investigate asymptotic spectral distribution of the combinatorial Laplacian (adjacency matrix) for the regular tree \({\mathcal {T}}_{b}^{(d)}\) of degree d with b-generation \((b\ge 3)\) in the vacuum and deformed vacuum states, we show that the vacuum spectral distributions, i.e., normalized Wigner semicircle laws, are the same as probability measures of the \({\mathcal {T}}_{b}^{(d)}\) in the weak case. We secondly get some expressions of quantum central limit theorems derived from quantum probability approaches for regular tree \({\mathcal {T}}_{b}^{(d)}\) . As an application, we finally calculate the probability amplitudes of continuous-time quantum random walks (CTQRWs, for short) on \({\mathcal {T}}_{b}^{(d)}\) by using Lanczos-based algorithm and spectral techniques for evaluation of this walk.
