Year: 2020
Type: article
Abstract: Colijn & Plazzotta ( Syst. Biol. 67:113-126, 2018) introduced a scheme for bijectively associating the unlabeled binary rooted trees with the positive integers. First, the rank 1 is associated with the 1-leaf tree. Proceeding recursively, ordered pair ( k 1 , k 2 ), k 1 ⩾ k 2 ⩾ 1, is then associated with the tree whose left subtree has rank k 1 and whose right subtree has rank k 2 . Following dictionary order on ordered pairs, the tree whose left and right subtrees have the ordered pair of ranks ( k 1 , k 2 ) is assigned rank k 1 ( k 1 − 1) / 2 + 1 + k 2 . With this ranking, given a number of leaves n , we determine recursions for a n , the smallest rank assigned to some tree with n leaves, and b n , the largest rank assigned to some tree with n leaves. For n equal to a power of 2, the value of a n is seen to increase exponentially with 2 α n for a constant α ≈ 1.24602; more generally, we show it is bounded a n < 1.5 n . The value of b n is seen to increase with for a constant β ≈ 1.05653. The great difference in the rates of increase for a n and b n indicates that as the index v is incremented, the number of leaves for the tree associated with rank v quickly traverses a wide range of values. We interpret the results in relation to applications in evolutionary biology. Mathematics subject classification 05C05, 92B10, 92D15
Source: Discrete Applied Mathematics
Author: Noah A. Rosenberg
Institution: Stanford University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $1,600
Awards: G4470145672, G9000023155, G9000096021
OpenAlex ID: W3114148850