On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees [article]

Noah A Rosenberg
2020 bioRxiv   pre-print
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 (k1,k2), k1 ≥ k2 ≥ 1, is then associated with the tree whose left subtree has rank k1 and whose right subtree has rank k2. Following dictionary order on ordered pairs, the tree whose left and right subtrees have the ordered pair of ranks (k1,k2) is
more » ... ssigned rank k1(k1-1)/2 + 1 + k2. 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 2α^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 2β^(2^n) 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.
doi:10.1101/2020.06.16.155184 fatcat:zl5lf43ryfaljc4n2spyden2ni