title: Leaf-to-leaf distances and their moments in finite and infinite ordered m-ary tree graphs.
abstract: We study the leaf-to-leaf distances on one-dimensionally ordered, full and complete m-ary tree graphs using a recursive approach. In our formulation, unlike in traditional graph theory approaches, leaves are ordered along a line emulating a one-dimensional lattice. We find explicit analytical formulas for the sum of all paths for arbitrary leaf separation r as well as the average distances and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.
authors.affiliation: Department of Physics and Centre for Scientific Computing, The University of Warwick, Coventry CV4 7AL, United Kingdom.
authors.forename: Andrew M
authors.initials: AM
authors.lastname: Goldsborough
authors.affiliation: Department of Physics and Centre for Scientific Computing, The University of Warwick, Coventry CV4 7AL, United Kingdom.
authors.forename: S Alex
authors.initials: SA
authors.lastname: Rautu
authors.affiliation: Department of Physics and Centre for Scientific Computing, The University of Warwick, Coventry CV4 7AL, United Kingdom.
authors.forename: Rudolf A
authors.initials: RA
authors.lastname: Römer