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. 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 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 2 β ( 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.

中文翻译：

---

关于未标记二叉树的 Colijn-Plazzotta 编号方案

Colijn & Plazzotta (Syst. Biol. 67:113-126, 2018) 介绍了一种将未标记的二叉树与正整数双射关联的方案。首先，秩 1 与一叶树相关联。递归地进行，有序对 (k 1, k 2)，k 1 ⩾ k 2 ⩾ 1，然后与左子树的秩为 k 1 且右子树的秩为 k 2 的树相关联。按照有序对的字典顺序，左子树和右子树具有有序秩对 (k 1, k 2) 的树被分配秩 k 1(k 1 - 1)/2 + 1 + k 2。通过这个排序，给定叶数 n，我们确定 a 的递归，它是分配给具有 n 个叶子的某棵树的最小等级，以及 bn ，分配给某些具有 n 个叶子的某棵树的最大等级。最小的秩an分配给最大平衡树，最大的秩 bn 被分配给毛毛虫。对于 n 等于 2 的幂，对于常数 α ≈ 1.24602，可以看到 an 的值随 2α n 呈指数增加；更一般地，我们证明它是有界的 < 1.5 n 。对于常数 β ≈ 1.05653，bn 的值随着 2 β ( 2 n ) 的增加而增加。an 和 bn 增长率的巨大差异表明，随着索引 v 的增加，与等级 v 相关的树的叶子数快速遍历广泛的值。我们解释了与进化生物学应用相关的结果。对于常数 β ≈ 1.05653，bn 的值随着 2 β ( 2 n ) 的增加而增加。an 和 bn 增长率的巨大差异表明，随着索引 v 的增加，与等级 v 相关的树的叶子数快速遍历广泛的值。我们解释了与进化生物学应用相关的结果。对于常数 β ≈ 1.05653，bn 的值随着 2 β ( 2 n ) 的增加而增加。an 和 bn 增长率的巨大差异表明，随着索引 v 的增加，与等级 v 相关的树的叶子数快速遍历广泛的值。我们解释了与进化生物学应用相关的结果。