  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., $D_{t}\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$) and a greater average
shortest path length $\langle\mathcal{W}_{t}\rangle$ (namely,
$\langle\mathcal{W}_{t}\rangle\sim|\mathcal{T}_{t}|^{\ln3/\ln5}$) where
$|\mathcal{T}_{t}|$ represents vertex number. Next, we determine Pearson
correlation coefficient and verify that networks $\mathcal{T}_{t}$ display
disassortative mixing structure. In addition, we 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 the analytic formula
for quantity $\langle\mathcal{H}_{t}\rangle$ as a function of vertex number
$|\mathcal{T}_{t}|$ shows a power-law form, i.e.,
$\langle\mathcal{H}_{t}\rangle\sim|\mathcal{T}_{t}|^{1+\ln3/\ln5}$.
Accordingly, we execute extensive experimental simulations, and demonstrate
that empirical analysis is in strong agreement with theoretical results.
Lastly, we provide a guide to extend the proposed iterative manner in order to
generate more general scale-free tree networks with large diameter.
