  Geodesic distance, commonly called shortest path length, has proved useful in
a great variety of disciplines. It has been playing a significant role in
search engine at present and so attracted considerable attention at the last
few decades, particularly, almost all data structures and corresponding
algorithms suitable to searching information generated based on treelike
models. Hence, we, in this paper, study in detail geodesic distance on some
treelike models which can be generated by three different types of operations,
including first-order subdivision, ($1,m$)-star-fractal operation and
$m$-vertex-operation. Compared to the most best used approaches for calculating
geodesic distance on graphs, for instance, enumeration method and matrix
multiplication, we take useful advantage of a novel method consisting in spirit
of the concept of vertex cover in the language of graph theory and mapping. For
each kind of treelike model addressed here, we certainly obtain an exact
solution for its geodesic distance using our method. With the help of computer
simulations, we confirm that the analytical results are in perfect agreement
with simulations. In addition, we also report some intriguing structure
properties on treelike models of two types among them. The one obeys
exponential degree distribution seen in many complex networks, by contrast, the
other possesses all but leaf vertices with identical degree and shows more
homogeneous topological structure than the former. Besides that, the both have,
in some sense, self-similar feature but instead the latter exhibits fractal
property.
