Summary: C. Colijn and G. Plazzotta [“A metric on phylogenetic tree shapes”, Syst. Biol. 67, 113–126 (2018; doi:10.1093/sysbio/syx046)] 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 \(( k_1 , k_2 )\), \(k_1 \geqslant k_2 \geqslant 1\), 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. The smallest rank \(a_n\) is assigned to the maximally balanced tree, and the largest rank \(b_n\) is assigned to the caterpillar. For \(n\) equal to a power of 2, the value of \(a_n\) is seen to increase exponentially with \(2 \alpha^n\) for a constant \(\alpha \approx 1 . 24602\); more generally, we show it is bounded \(a_n < 1 . 5^n\). The value of \(b_n\) is seen to increase with \(2 \beta^{( 2^n )}\) for a constant \(\beta \approx 1 . 05653\). 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.