A note on growing binary trees

The class @? of binary search trees is studied. A leaf is a vertex of degree 0; @?"n is the subset of @? consisting of trees with n leaves. We grow trees in @?"n from @?"n - 1 thereby inducing a probability measure on @?"n. We will show that the expected value of the average leaf distance of t @? @?"n is asymptotic to log"2n as n -> ~.