  Given a tree T, one can define the local mean at some subtree S to be the
average order of subtrees containing S. It is natural to ask which subtree of
order k achieves the maximal/minimal local mean among all the subtrees of the
same order and what properties it has. We call such subtrees k-maximal
subtrees. Wagner and Wang showed in 2016 that a 1- maximal subtree is a vertex
of degree 1 or 2. This paper shows that for any integer k = 1, . . . , |T| , a
k-maximal subtree has at most one leaf whose degree is greater than 2 and at
least one leaf whose degree is at most 2. Furthermore, we show that a k-maximal
subtree has a leaf of degree greater than 2 only when all its other leaves are
leaves in T as well. In the second part, this paper introduces the local
density as a normalization of local means, for the sake of comparing subtrees
of different orders, and shows that the local density at subtree S is
lower-bounded by 1/2 with equality if and only if S contains the core of T. On
the other hand, local density can be arbitrarily close to 1.
