  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.
