When Sparse Trees Are Dense and When Trees Are Indistinguishable

We will consider two questions proposed by R. Jamison in 1983 regarding the average subtree order of trees. The first question asks if it is true that a tree with internal vertex degree at least 3 has its average subtree order being at least half of the total number of vertices. We provide a positive answer to this question and some further observations on how the trees with large or small average subtree order look like. The second question asks if every two non-isomorphic trees of the same order have different average subtree orders. We provide an example showing that the answer is negative. This example follows from a generalization of Schwenk's classical result on the cospectral mate of a tree.