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

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 assigned rank k1(k1 - 1)/2 + 1 + k2. With this ranking, given a number of leaves n, we determine recursions for an, the smallest rank assigned to some tree with n leaves, and bn, the largest rank assigned to some tree with n leaves. For n equal to a power of 2, the value of an is seen to increase exponentially with 2n for a constant {approx} 1.24602; more generally, we show it is bounded an < 1.5n. The value of bn is seen to increase with [Formula] for a constant {beta} {approx} 1.05653. The great difference in the rates of increase for an and bn 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 classification05C05, 92B10, 92D15