Ranking and unranking trees with a given number or a given set of leaves

TL;DR

This paper introduces algorithms for ranking and unranking Cayley trees with specified leaf sets or fixed leaf counts, enabling efficient enumeration and reconstruction of such trees.

Contribution

It provides the first efficient algorithms for ranking and unranking degree-restricted Cayley trees based on leaf constraints.

Findings

Algorithms operate in O(n^2) comparisons and O(n) arithmetic operations.

Number of Cayley trees with k leaves grows roughly as n! for fixed k.

Algorithms handle large numbers with length O(nlog(n)).

Abstract

In this paper, we provide algorithms to rank and unrank certain degree-restricted classes of Cayley trees (spanning trees of the n-vertex complete graph). Specifically, we consider classes of trees that have a given set of leaves or a fixed number k of leaves. For fixed k, the number of Cayley trees with n vertices and k leaves grows roughly as n! and hence the ranks have O(nlog_2(n)) bits. Our ranking and unranking algorithms require at most O(n^2) comparisons of numbers less than or equal to n plus O(n) operations of multiplication, division, addition, substraction and comparision on numbers of length O(nlog(n)).