A note on the largest number of red nodes in red-black trees

TL;DR

This paper investigates the maximum number of red nodes in red-black trees, developing efficient algorithms and a closed-form formula to compute this maximum for any number of keys.

Contribution

It introduces improved algorithms and derives a closed-form expression for the maximum red nodes in red-black trees, advancing understanding of their structure.

Findings

Developed an $O(n^2 \log n)$ dynamic programming solution.

Created $O(\log n)$ time recursive and nonrecursive algorithms.

Derived a closed-form formula for the maximum number of red nodes.

Abstract

In this paper, we are interested in the number of red nodes in red-black trees. We first present an $O(n^2\log n)$ time dynamic programming solution for computing $r(n)$, the largest number of red internal nodes in a red-black tree on $n$ keys. Then the algorithm is improved to some $O(\log n)$ time recursive and nonrecursive algorithms. Based on these improved algorithms we finally find a closed-form solution of $r(n)$.