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

A Method for Geodesic Distance on Subdivision of Trees with Arbitrary Orders and Their Applications