On the maximum mean subtree order of trees

A subtree of a tree is any induced subgraph that again (i.e., connected). The mean order the average number vertices its subtrees. This invariant was first analyzed in 1980s by Jamison. An intriguing open question raised Jamison asks whether maximum order, given tree, always attained some caterpillar. While we do not completely resolve this conjecture, find evidence favor proving different features trees attain maximum. For example, show diameter n with must be very close to n. Moreover, equal n−2log2n+O(1). local which all subtrees containing fixed vertex, can even more precise: broom and it n−log2n+O(1).