Subtractive random forests

Abstract

Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$ are attached sequentially such that the parent of vertex $n$ is $n-Z_n$, where the $Z_n$ are i.i.d.\ random variables taking values in $\mathbb N$. We study several characteristics of the resulting random forest. In particular, we establish bounds for the expected tree sizes, the number of trees in the forest, the number of leaves, the maximum degree, and the height of the forest. We show that for all distributions of the $Z_n$, the forest contains at most one infinite tree, almost surely. If ${\mathbb E} Z_n < \infty$, then there is a unique infinite tree and the total size of the remaining trees is finite, with finite expected value if ${\mathbb E}Z_n^2 < \infty$. If ${\mathbb E} Z_n = \infty$ then almost surely all trees are finite.