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 $C_{\varphi }$ 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 $C_{\varphi }$ on ${\mathbb{T}}_p$ and finally prove there are no compact composition operators on ${\mathbb{T}}_{p,0}$ spaces.