  The problem of how to estimate diffusion on a graph effectively is of
importance both theoretically and practically. In this paper, we make use of
two widely studied indices, geodesic distance and mean first-passage time
($MFPT$) for random walk, to consider such a problem on some treelike models of
interest. To this end, we first introduce several types of operations, for
instance, $m$th-order subdivision and ($1,m$)-star-fractal operation, to
generate the potential candidate models. And then, we develop a class of novel
techniques based on mapping for calculating the exact formulas of the both
quantities above on our models. Compared to those previous tools including
matrix-based methods for addressing the issue of this type, the techniques
proposed here are more general mainly because we generalize the initial
condition for creating these models. Meantime, in order to show applications of
our techniques to many other treelike models, the two popularly discussed
models, Cayley tree $C(t,n)$ and Exponential tree $\mathcal{T}(t;m)$, are
chosen to serve as representatives. While their correspondence solutions have
been reported, our techniques are more convenient to implement than those
pre-existing ones and thus this certifies our statements strongly. Final, we
distinguish the difference among results and attempt to make some heuristic
explanations to the reasons why the phenomena appear.
