  Scale-free networks are prevalently observed in a great variety of complex
systems, which triggers various researches relevant to networked models of such
type. In this work, we propose a family of growth tree networks
$\mathcal{T}_{t}$, which turn out to be scale-free, in an iterative manner. As
opposed to most of published tree models with scale-free feature, our tree
networks have the power-law exponent $\gamma=1+\ln5/\ln2$ that is obviously
larger than $3$. At the same time, "small-world" property can not be found
particularly because models $\mathcal{T}_{t}$ have an ultra-large diameter
$D_{t}$, i.e., $D_{t}\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$ where
$|\mathcal{T}_{t}|$ represents vertex number. In addition, we also study random
walks on tree networks $\mathcal{T}_{t}$ and derive exact solution to mean
hitting time $\langle\mathcal{H}_{t}\rangle$. The results suggest that analytic
formula for quantity $\langle\mathcal{H}_{t}\rangle$ as a function of vertex
number $\mathcal{T}_{t}$ has a power-law form, i.e.,
$\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln3/\ln5}$ , a
characteristic that is rarely encountered in the realm of scale-free tree
networks. Lastly, we execute extensive experimental simulations, and find that
empirical analysis is in strong agreement with theoretical results.
