Broutin, NicolasDevroye, LucLugosi, GáborOliveira, Roberto Imbuzeiro2025-06-172025-06-172024Broutin N, Devroye L, Lugosi G, Oliveira RI. Subtractive random forests. ALEA, Lat. Am. J. Probab. Math. Stat. 2024;21:575-91. DOI: 10.30757/ALEA.v21-231980-0436http://hdl.handle.net/10230/70706Motivated 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 ≤ 0 is the root of a tree. Vertices labeled by positive integers n ≥ 1 are attached sequentially such that the parent of vertex n is n−Zn, where the Zn are i.i.d. random variables taking values in Z+. 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 Zn, the forest contains at most one infinite tree, almost surely. If EZn < ∞, then there is a unique infinite tree and the total size of the remaining trees is finite, with finite expected value if EZ2 n < ∞. If EZn = ∞ then almost surely all trees are finite.application/pdfengALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.info:eu-repo/semantics/article2025-06-17http://dx.doi.org/10.30757/ALEA.v21-23Subtractive random forestsRecommendation systemsAncestral processinfo:eu-repo/semantics/openAccess