Average distance in growing trees

Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barab'asi–Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to $m=1$ nodes. Average node-node distance $d$ is calculated numerically in evolving trees as dependent on the number of nodes $N$. The results for $N$ not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance $d$ for large $N$ can be approximated by $d=2\ln (N)+c_1$ for the exponential trees, and $d=\ln (N)+c_2$ for the scale-free trees, where the $c_i$ are constant. We derive also iterative equations for $d$ and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.