The Power of Choice in the Construction of Recursive Trees

Abstract

The power of choice is known to change the character of random structures and produce desirable optimization effects. We discuss generalizations of random recursive trees, grown under the choice to meet optimization criteria. Specifically, we discuss the random k-minimal (k-maximal) label recursive tree, where a set of k candidate parents, instead of one as in the usual recursive tree, is selected and the node with minimal (maximal) label among them is assigned as parent for the next node. These models are proposed as alternatives for D’Souza et al. (Eur Phys J B59:535–543, 2007) minimal and maximal depth models. The advantage of the label models is that they are tractable and at the same time provide approximations and bounds for the depth models. For the depth of nodes in label models we give the average behavior and exact distributions involving Stirling’s numbers and derive Gaussian limit laws.