# On the centroid of increasing trees

## TL;DR

This paper extends known results about centroid nodes from random recursive trees to a broader class of increasing trees, analyzing their distribution, depth, label, and subtree size.

## Contribution

It generalizes the behavior of centroid nodes to simple increasing trees, including plane-oriented and d-ary increasing trees, with new distribution and moment results.

## Key findings

- Limits of distributions for centroid depth and label
- Distribution of subtree sizes at the centroid
- Analytical expressions for moments of centroid properties

## Abstract

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 $d$-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.