A335729 Number of "coprime" pairs of binary trees with n carets (see comments).

1, 2, 10, 68, 546, 4872, 46782, 474180, 5010456, 54721224, 613912182, 7042779996, 82329308040, 978034001472

Offset: 1

Author

Dennis Sweeney, Jul 17 2020

Keywords

• nonn
• more

Comments

a(n) is the number of ordered pairs of rooted binary trees (with all nodes having either 2 or 0 ordered children) each with n non-leaf nodes (sometimes called carets) such that the pair is "coprime".

Call such a tree-pair (A, B) coprime if, upon labeling the leaves 1 through n + 1 (left to right), there does not exist a non-leaf, non-root node a of A and a non-leaf, non-root node b of B such that the set of labels on the descendant leaves of a equals the set of labels on the descendant leaves of b, i.e., if A and B have no proper subtrees "in the same place".

Examples

			A coprime tree-pair with 5 carets:
       .              .
      / \            / \
     / \ \          /   \
    / / \ \        / \   \
   / / \ \ \      / \ \   \
  / / \ \ \ \    / \ \ \ / \
  1 2 3 4 5 6    1 2 3 4 5 6
A non-coprime tree-pair (both have a subtree on leaves 1-2-3-4):
       .              .
      / \            / \
     / \ \          /   \
    / \ \ \        / \   \
   /   \ \ \      / / \   \
  / \ / \ \ \    / / \ \ / \
  1 2 3 4 5 6    1 2 3 4 5 6
Below we will represent a binary tree by a bracketing of the leaf labels 1 through n + 1 (a vertex of an associahedron). A tree is represented by a balanced string, and its left and right child subtrees are represented by two maximal balanced proper substrings, in order.
For n = 2, the a(2) = 2 coprime tree-pairs are:
  ([[12]3], [1[23]]),
  ([1[23]], [[12]3]).
For n = 3, the a(3) = 10 coprime tree-pairs are:
  ([1[2[34]]], [[1[23]]4]),
  ([1[2[34]]], [[[12]3]4]),
  ([1[[23]4]], [[12][34]]),
  ([1[[23]4]], [[[12]3]4]),
  ([[12][34]], [1[[23]4]]),
  ([[12][34]], [[1[23]]4]),
  ([[1[23]]4], [1[2[34]]]),
  ([[1[23]]4], [[12][34]]),
  ([[[12]3]4], [1[2[34]]]),
  ([[[12]3]4], [1[[23]4]]).
		

Links

• S. Cleary and R. Maio, Counting difficult tree pairs with respect to the rotation distance problem, arXiv:2001.06407 [cs.DS], 2020.

Crossrefs

a(n) counts a subset of the tree-pairs that A111713 counts; "coprime" is a stronger condition than "reduced". It appears that for n > 1, a(n)/2 coincides with A257887.