Profiles of random trees: correlation and width of random recursive trees and binary search trees

Abstract

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp signchanges when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

Profiles of random trees: Plane‐oriented recursive trees

“…2, highlighting in particular the discontinuous sign-change at 1 2 . Intuitive interpretations of the results are similar to those provided in [18] for recursive trees.…”

“…Then (17) is derived, similarly as in [18], by a uniform estimate for the function on the righthand side in the u, v plane (by applying the singularity analysis of Flajolet and Odlyzko [21]), and then by extending the saddle point method used in [26].…”

“…The dominant term on the right-hand side is indeed tight; the approach used in [18] based on correlations of two level sizes can be applied to show that…”

Structural Transitions in Densifying Networks

“…As we will show, copying leads to highly non-trivial networks, but the simplicity of this mechanism allows for analytical solution for many network properties. When p = 0, a network built by copying is a random recursive tree [38][39][40], while for p = 1, a complete graph arises if the initial graph is also complete. For p < 1 2 , the network is sparse, while for p ≥ 1 2 , the number of links grows superlinearly with N and the network is dense.…”

The Subtree Size Profile of Plane-oriented Recursive Trees

“…Several recent studies have been concerned with the node profile of rooted random trees, where the node profile is defined as the number of nodes at distance k from the root; for random binary search trees and recursive trees see Chauvin et al [4], Chauvin et al [5], Drmota and Hwang [11], [12], Fuchs et al [22]; for random plane-oriented recursive trees see Hwang [23]; for other types of random trees see Drmota and Gittenberger [10], Drmota et al [13], Drmota and Szpankowski [14], Park et al [25].…”