  Over some types of trees with a given number of vertices, which trees
minimize or maximize the total number of subtrees or leaf containing subtrees
are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the
total number of subtrees (resp. leaf containing subtrees) among $n$-vertex
trees with a given matching number is determined; as a consequence, the
$n$-vertex tree with domination number $\gamma$ maximizing the total number of
subtrees (resp. leaf containing subtrees) is characterized. (2)\, Sharp lower
bound on the total number of leaf containing subtrees among $n$-vertex trees
with maximum degree at least $\Delta$ is determined; as a consequence the
$n$-vertex tree with maximum degree at least $\Delta$ having a perfect matching
minimizing the total number of subtrees (resp. leaf containing subtrees) is
characterized. (3)\, Sharp upper (resp. lower) bound on the total number of
leaf containing subtrees among the set of all $n$-vertex trees with $k$ leaves
(resp. the set of all $n$-vertex trees of diameter $d$) is determined.
