Composition operators on Hardy spaces of the homogenous rooted trees

In \cite{CO-Tp-spaces}, the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space $\mathbb{T}_{p}$ defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator $C_\phi$ to be bounded on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces. Also, we discuss the compactness of $C_\phi$ on $\mathbb{T}_{p}$ and finally prove there are no compact composition operators on $\mathbb{T}_{p,0}$ spaces.