Exponentially preferential trees

We introduce the exponentially preferential recursive tree and study some properties related to the degree profile of nodes in the tree. The definition of the tree involves a radix $a\gt 0$ . In a tree of size $n$ (nodes), the nodes are labeled with the numbers $1,2, \ldots ,n$ . The node labeled $i$ attracts the future entrant $n+1$ with probability proportional to $a^i$ . We dedicate an early section for algorithms to generate and visualize the trees in different regimes. We study the asymptotic distribution of the outdegree of node $i$ , as $n\to \infty$ , and find three regimes according to whether $0 \lt a \lt 1$ (subcritical regime), $a=1$ (critical regime), or $a\gt 1$ (supercritical regime). Within any regime, there are also phases depending on a delicate interplay between $i$ and $n$ , ramifying the asymptotic distribution within the regime into “early,” “intermediate” and “late” phases. In certain phases of certain regimes, we find asymptotic Gaussian laws. In certain phases of some other regimes, small oscillations in the asymototic laws are detected by the Poisson approximation techniques.