  In this paper, we investigate random walks in a family of small-world trees
having an exponential degree distribution. First, we address a trapping
problem, that is, a particular case of random walks with an immobile trap
located at the initial node. We obtain the exact mean trapping time defined as
the average of first-passage time (FPT) from all nodes to the trap, which
scales linearly with the network order $N$ in large networks. Then, we
determine analytically the mean sending time, which is the mean of the FPTs
from the initial node to all other nodes, and show that it grows with $N$ in
the order of $N \ln N$. After that, we compute the precise global mean
first-passage time among all pairs of nodes and find that it also varies in the
order of $N \ln N$ in the large limit of $N$. After obtaining the relevant
quantities, we compare them with each other and related our results to the
efficiency for information transmission by regarding the walker as an
information messenger. Finally, we compare our results with those previously
reported for other trees with different structural properties (e.g., degree
distribution), such as the standard fractal trees and the scale-free
small-world trees, and show that the shortest path between a pair of nodes in a
tree is responsible for the scaling of FPT between the two nodes.
