4 research outputs found
Additive tree functionals with small toll functions and subtrees of random trees
Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size decreases exponentially with ). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves
On the centroid of increasing trees
A centroid node in a tree is a node for which the sum of the distances to all
other nodes attains its minimum, or equivalently a node with the property that
none of its branches contains more than half of the other nodes. We generalise
some known results regarding the behaviour of centroid nodes in random
recursive trees (due to Moon) to the class of very simple increasing trees,
which also includes the families of plane-oriented and -ary increasing
trees. In particular, we derive limits of distributions and moments for the
depth and label of the centroid node nearest to the root, as well as for the
size of the subtree rooted at this node