In this work, we propose a principled framework using Vertex-based and Edge-based uniform generation mechanisms to build stochastic uniform growth tree networks that have a wide range of applications in various fields including physics, engineering, chemistry, ect., and then uncover the associated structural features analytically. When considering vertex-degree distribution, there exist three different classes of forms in the thermodynamic limit, i.e., exponential distribution, power-law distribution along with multiple-point distribution. At meantime, three distinct structural shapes are observed in the study of fractal phenomena, that is, fractal feature, critical phenomenon and non-fractal property. In addition, we obtain the analytical solution to fractal dimension for fractal structure from the probability point of view. More importantly, some well-known models, for instance, Vicsek fractal and T-graph, fall into our framework. Next, we precisely consider two families of stochastic uniform growth tree networks generated through the proposed framework. Specifically, we derive the analytic solution to mean hitting time$\langle \mathcal {H}\rangle$for measuring efficiency of delivering information on networks in a random-walk-based manner, and find that the introduction of randomness certainly enriches the scaling exponent of quantity$\langle \mathcal {H}\rangle$. Finally, we conduct extensive experiments, which suggests that computer simulations are in good agreement with theoretical analysis.

Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree$\mathcal {T}$using two types of well-known operations, first-order subdivision and ($1,m$)-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth trees using a single edge as seed in the literature, we propose a novel method for deriving closed-form solutions on the presented trees completely. Meanwhile, our technique is more general and convenient to implement compared to those previous methods mainly because there are not complicated calculations needed. In addition, the closed-form expression of mean first-passage time ($MFPT$) for random walk on each member in tree families is also readily obtained according to connection of our obtained results to effective resistance of corresponding electric networks. The results suggest that the two topological operations above are sharply different from each other due to$MFPT$for random walks, and, however, have likely to show the similar performance, at least, on geodesic distance.