Trees with given degree sequences that have minimal subtrees

In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $\pi =(d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_5,1,\mathrm{.}\mathrm{.}\mathrm{.},1)$ have been characterized.