A265279 Number of pairs of binary trees with n leaves which have only a single common parse word under the grammar G = {0 -> 12, 0 -> 21, 1 -> 02, 1 -> 20, 2 -> 01, 2 -> 10}.

1, 5, 30, 196, 1348, 9603, 70210, 523732, 3969948, 30490434

Offset

3,2

Comments

We say that a word w in {0,1,2}^* is a parse word of a binary tree T if T is the derivation tree of w under the given grammar G.

Parse words are counted only up to permutation of the three letters.

The statement that all pairs of binary trees have at least one common parse word under G is equivalent to the four-color theorem.

Links

Table of n, a(n) for n=3..12.

B. Cooper and E. Rowland and D. Zeilberger, Toward a language theoretic proof of the four color theorem, arXiv:1006.1324 [math.CO], 2011-2012.

B. Cooper and E. Rowland and D. Zeilberger, Toward a language theoretic proof of the four color theorem, Advances in Applied Mathematics 48 (2012), 414-431.

E. Rowland, Mathematica program ParseWords.m; Local copy

D. Zeilberger, Maple program LOU.txt; Local copy

Example

There are exactly two binary trees with n = 3 leaves, the first one parses the words 101 and 110, the second one parses the words 011 and 101 (and all words obtained by permuting the letters 0, 1, 2). Thus this single pair of trees has precisely one common parse word.

Maple

read `LOU.txt`;  # see link above
seq(nops(MinLou(n)[2]), n=3..7);

Mathematica

BinaryTrees[1] = {{}};
BinaryTrees[n_Integer /; n > 1] := BinaryTrees[n] =
  Join @@ Table[Tuples[{BinaryTrees[i], BinaryTrees[n - i]}], {i, n - 1}];
ParseWords[{}, root_] := {{root}};
ParseWords[tree_List, root_: 0] := ParseWords[tree, root] =
  With[{col = DeleteCases[{0, 1, 2}, root]}, Join[
    Flatten /@ Tuples[{ParseWords[tree[[1]], col[[1]]], ParseWords[tree[[2]], col[[2]]]}],
    Flatten /@ Tuples[{ParseWords[tree[[1]], col[[2]]], ParseWords[tree[[2]], col[[1]]]}]
  ]];
MinTreePairs[n_Integer /; n > 0] :=
  Count[Length[Intersection[ParseWords[#1], ParseWords[#2]]]& @@@ Subsets[BinaryTrees[n], {2}], 2];

Crossrefs

Sequence in context: A158828 A264910 A196471 * A034164 A322257 A103433

Adjacent sequences: A265276 A265277 A265278 * A265280 A265281 A265282

Keyword

nonn,hard,more

Author

Christoph Koutschan, Apr 06 2016

Status

approved