edit this statistic or download as text // json
Identifier
Values
=>
Cc0010;cc-rep
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[[[.,.],.],[[[.,.],[.,.]],.]]=>2 [[[.,.],.],[[[.,[.,.]],.],.]]=>1 [[[.,.],.],[[[[.,.],.],.],.]]=>1 [[.,[.,[.,.]]],[.,[.,[.,.]]]]=>1 [[.,[.,[.,.]]],[.,[[.,.],.]]]=>1 [[.,[.,[.,.]]],[[.,.],[.,.]]]=>2 [[.,[.,[.,.]]],[[.,[.,.]],.]]=>1 [[.,[.,[.,.]]],[[[.,.],.],.]]=>1 [[.,[[.,.],.]],[.,[.,[.,.]]]]=>1 [[.,[[.,.],.]],[.,[[.,.],.]]]=>1 [[.,[[.,.],.]],[[.,.],[.,.]]]=>2 [[.,[[.,.],.]],[[.,[.,.]],.]]=>1 [[.,[[.,.],.]],[[[.,.],.],.]]=>1 [[[.,.],[.,.]],[.,[.,[.,.]]]]=>2 [[[.,.],[.,.]],[.,[[.,.],.]]]=>2 [[[.,.],[.,.]],[[.,.],[.,.]]]=>3 [[[.,.],[.,.]],[[.,[.,.]],.]]=>2 [[[.,.],[.,.]],[[[.,.],.],.]]=>2 [[[.,[.,.]],.],[.,[.,[.,.]]]]=>1 [[[.,[.,.]],.],[.,[[.,.],.]]]=>1 [[[.,[.,.]],.],[[.,.],[.,.]]]=>2 [[[.,[.,.]],.],[[.,[.,.]],.]]=>1 [[[.,[.,.]],.],[[[.,.],.],.]]=>1 [[[[.,.],.],.],[.,[.,[.,.]]]]=>1 [[[[.,.],.],.],[.,[[.,.],.]]]=>1 [[[[.,.],.],.],[[.,.],[.,.]]]=>2 [[[[.,.],.],.],[[.,[.,.]],.]]=>1 [[[[.,.],.],.],[[[.,.],.],.]]=>1 [[.,[.,[.,[.,.]]]],[.,[.,.]]]=>1 [[.,[.,[.,[.,.]]]],[[.,.],.]]=>1 [[.,[.,[[.,.],.]]],[.,[.,.]]]=>1 [[.,[.,[[.,.],.]]],[[.,.],.]]=>1 [[.,[[.,.],[.,.]]],[.,[.,.]]]=>2 [[.,[[.,.],[.,.]]],[[.,.],.]]=>2 [[.,[[.,[.,.]],.]],[.,[.,.]]]=>1 [[.,[[.,[.,.]],.]],[[.,.],.]]=>1 [[.,[[[.,.],.],.]],[.,[.,.]]]=>1 [[.,[[[.,.],.],.]],[[.,.],.]]=>1 [[[.,.],[.,[.,.]]],[.,[.,.]]]=>2 [[[.,.],[.,[.,.]]],[[.,.],.]]=>2 [[[.,.],[[.,.],.]],[.,[.,.]]]=>2 [[[.,.],[[.,.],.]],[[.,.],.]]=>2 [[[.,[.,.]],[.,.]],[.,[.,.]]]=>2 [[[.,[.,.]],[.,.]],[[.,.],.]]=>2 [[[[.,.],.],[.,.]],[.,[.,.]]]=>2 [[[[.,.],.],[.,.]],[[.,.],.]]=>2 [[[.,[.,[.,.]]],.],[.,[.,.]]]=>1 [[[.,[.,[.,.]]],.],[[.,.],.]]=>1 [[[.,[[.,.],.]],.],[.,[.,.]]]=>1 [[[.,[[.,.],.]],.],[[.,.],.]]=>1 [[[[.,.],[.,.]],.],[.,[.,.]]]=>2 [[[[.,.],[.,.]],.],[[.,.],.]]=>2 [[[[.,[.,.]],.],.],[.,[.,.]]]=>1 [[[[.,[.,.]],.],.],[[.,.],.]]=>1 [[[[[.,.],.],.],.],[.,[.,.]]]=>1 [[[[[.,.],.],.],.],[[.,.],.]]=>1 [[.,[.,[.,[.,[.,.]]]]],[.,.]]=>1 [[.,[.,[.,[[.,.],.]]]],[.,.]]=>1 [[.,[.,[[.,.],[.,.]]]],[.,.]]=>2 [[.,[.,[[.,[.,.]],.]]],[.,.]]=>1 [[.,[.,[[[.,.],.],.]]],[.,.]]=>1 [[.,[[.,.],[.,[.,.]]]],[.,.]]=>2 [[.,[[.,.],[[.,.],.]]],[.,.]]=>2 [[.,[[.,[.,.]],[.,.]]],[.,.]]=>2 [[.,[[[.,.],.],[.,.]]],[.,.]]=>2 [[.,[[.,[.,[.,.]]],.]],[.,.]]=>1 [[.,[[.,[[.,.],.]],.]],[.,.]]=>1 [[.,[[[.,.],[.,.]],.]],[.,.]]=>2 [[.,[[[.,[.,.]],.],.]],[.,.]]=>1 [[.,[[[[.,.],.],.],.]],[.,.]]=>1 [[[.,.],[.,[.,[.,.]]]],[.,.]]=>2 [[[.,.],[.,[[.,.],.]]],[.,.]]=>2 [[[.,.],[[.,.],[.,.]]],[.,.]]=>3 [[[.,.],[[.,[.,.]],.]],[.,.]]=>2 [[[.,.],[[[.,.],.],.]],[.,.]]=>2 [[[.,[.,.]],[.,[.,.]]],[.,.]]=>2 [[[.,[.,.]],[[.,.],.]],[.,.]]=>2 [[[[.,.],.],[.,[.,.]]],[.,.]]=>2 [[[[.,.],.],[[.,.],.]],[.,.]]=>2 [[[.,[.,[.,.]]],[.,.]],[.,.]]=>2 [[[.,[[.,.],.]],[.,.]],[.,.]]=>2 [[[[.,.],[.,.]],[.,.]],[.,.]]=>3 [[[[.,[.,.]],.],[.,.]],[.,.]]=>2 [[[[[.,.],.],.],[.,.]],[.,.]]=>2 [[[.,[.,[.,[.,.]]]],.],[.,.]]=>1 [[[.,[.,[[.,.],.]]],.],[.,.]]=>1 [[[.,[[.,.],[.,.]]],.],[.,.]]=>2 [[[.,[[.,[.,.]],.]],.],[.,.]]=>1 [[[.,[[[.,.],.],.]],.],[.,.]]=>1 [[[[.,.],[.,[.,.]]],.],[.,.]]=>2 [[[[.,.],[[.,.],.]],.],[.,.]]=>2 [[[[.,[.,.]],[.,.]],.],[.,.]]=>2 [[[[[.,.],.],[.,.]],.],[.,.]]=>2 [[[[.,[.,[.,.]]],.],.],[.,.]]=>1 [[[[.,[[.,.],.]],.],.],[.,.]]=>1 [[[[[.,.],[.,.]],.],.],[.,.]]=>2 [[[[[.,[.,.]],.],.],.],[.,.]]=>1 [[[[[[.,.],.],.],.],.],[.,.]]=>1 [[.,[.,[.,[.,[.,[.,.]]]]]],.]=>0 [[.,[.,[.,[.,[[.,.],.]]]]],.]=>0 [[.,[.,[.,[[.,.],[.,.]]]]],.]=>1 [[.,[.,[.,[[.,[.,.]],.]]]],.]=>0 [[.,[.,[.,[[[.,.],.],.]]]],.]=>0 [[.,[.,[[.,.],[.,[.,.]]]]],.]=>1 [[.,[.,[[.,.],[[.,.],.]]]],.]=>1 [[.,[.,[[.,[.,.]],[.,.]]]],.]=>1 [[.,[.,[[[.,.],.],[.,.]]]],.]=>1 [[.,[.,[[.,[.,[.,.]]],.]]],.]=>0 [[.,[.,[[.,[[.,.],.]],.]]],.]=>0 [[.,[.,[[[.,.],[.,.]],.]]],.]=>1 [[.,[.,[[[.,[.,.]],.],.]]],.]=>0 [[.,[.,[[[[.,.],.],.],.]]],.]=>0 [[.,[[.,.],[.,[.,[.,.]]]]],.]=>1 [[.,[[.,.],[.,[[.,.],.]]]],.]=>1 [[.,[[.,.],[[.,.],[.,.]]]],.]=>2 [[.,[[.,.],[[.,[.,.]],.]]],.]=>1 [[.,[[.,.],[[[.,.],.],.]]],.]=>1 [[.,[[.,[.,.]],[.,[.,.]]]],.]=>1 [[.,[[.,[.,.]],[[.,.],.]]],.]=>1 [[.,[[[.,.],.],[.,[.,.]]]],.]=>1 [[.,[[[.,.],.],[[.,.],.]]],.]=>1 [[.,[[.,[.,[.,.]]],[.,.]]],.]=>1 [[.,[[.,[[.,.],.]],[.,.]]],.]=>1 [[.,[[[.,.],[.,.]],[.,.]]],.]=>2 [[.,[[[.,[.,.]],.],[.,.]]],.]=>1 [[.,[[[[.,.],.],.],[.,.]]],.]=>1 [[.,[[.,[.,[.,[.,.]]]],.]],.]=>0 [[.,[[.,[.,[[.,.],.]]],.]],.]=>0 [[.,[[.,[[.,.],[.,.]]],.]],.]=>1 [[.,[[.,[[.,[.,.]],.]],.]],.]=>0 [[.,[[.,[[[.,.],.],.]],.]],.]=>0 [[.,[[[.,.],[.,[.,.]]],.]],.]=>1 [[.,[[[.,.],[[.,.],.]],.]],.]=>1 [[.,[[[.,[.,.]],[.,.]],.]],.]=>1 [[.,[[[[.,.],.],[.,.]],.]],.]=>1 [[.,[[[.,[.,[.,.]]],.],.]],.]=>0 [[.,[[[.,[[.,.],.]],.],.]],.]=>0 [[.,[[[[.,.],[.,.]],.],.]],.]=>1 [[.,[[[[.,[.,.]],.],.],.]],.]=>0 [[.,[[[[[.,.],.],.],.],.]],.]=>0 [[[.,.],[.,[.,[.,[.,.]]]]],.]=>1 [[[.,.],[.,[.,[[.,.],.]]]],.]=>1 [[[.,.],[.,[[.,.],[.,.]]]],.]=>2 [[[.,.],[.,[[.,[.,.]],.]]],.]=>1 [[[.,.],[.,[[[.,.],.],.]]],.]=>1 [[[.,.],[[.,.],[.,[.,.]]]],.]=>2 [[[.,.],[[.,.],[[.,.],.]]],.]=>2 [[[.,.],[[.,[.,.]],[.,.]]],.]=>2 [[[.,.],[[[.,.],.],[.,.]]],.]=>2 [[[.,.],[[.,[.,[.,.]]],.]],.]=>1 [[[.,.],[[.,[[.,.],.]],.]],.]=>1 [[[.,.],[[[.,.],[.,.]],.]],.]=>2 [[[.,.],[[[.,[.,.]],.],.]],.]=>1 [[[.,.],[[[[.,.],.],.],.]],.]=>1 [[[.,[.,.]],[.,[.,[.,.]]]],.]=>1 [[[.,[.,.]],[.,[[.,.],.]]],.]=>1 [[[.,[.,.]],[[.,.],[.,.]]],.]=>2 [[[.,[.,.]],[[.,[.,.]],.]],.]=>1 [[[.,[.,.]],[[[.,.],.],.]],.]=>1 [[[[.,.],.],[.,[.,[.,.]]]],.]=>1 [[[[.,.],.],[.,[[.,.],.]]],.]=>1 [[[[.,.],.],[[.,.],[.,.]]],.]=>2 [[[[.,.],.],[[.,[.,.]],.]],.]=>1 [[[[.,.],.],[[[.,.],.],.]],.]=>1 [[[.,[.,[.,.]]],[.,[.,.]]],.]=>1 [[[.,[.,[.,.]]],[[.,.],.]],.]=>1 [[[.,[[.,.],.]],[.,[.,.]]],.]=>1 [[[.,[[.,.],.]],[[.,.],.]],.]=>1 [[[[.,.],[.,.]],[.,[.,.]]],.]=>2 [[[[.,.],[.,.]],[[.,.],.]],.]=>2 [[[[.,[.,.]],.],[.,[.,.]]],.]=>1 [[[[.,[.,.]],.],[[.,.],.]],.]=>1 [[[[[.,.],.],.],[.,[.,.]]],.]=>1 [[[[[.,.],.],.],[[.,.],.]],.]=>1 [[[.,[.,[.,[.,.]]]],[.,.]],.]=>1 [[[.,[.,[[.,.],.]]],[.,.]],.]=>1 [[[.,[[.,.],[.,.]]],[.,.]],.]=>2 [[[.,[[.,[.,.]],.]],[.,.]],.]=>1 [[[.,[[[.,.],.],.]],[.,.]],.]=>1 [[[[.,.],[.,[.,.]]],[.,.]],.]=>2 [[[[.,.],[[.,.],.]],[.,.]],.]=>2 [[[[.,[.,.]],[.,.]],[.,.]],.]=>2 [[[[[.,.],.],[.,.]],[.,.]],.]=>2 [[[[.,[.,[.,.]]],.],[.,.]],.]=>1 [[[[.,[[.,.],.]],.],[.,.]],.]=>1 [[[[[.,.],[.,.]],.],[.,.]],.]=>2 [[[[[.,[.,.]],.],.],[.,.]],.]=>1 [[[[[[.,.],.],.],.],[.,.]],.]=>1 [[[.,[.,[.,[.,[.,.]]]]],.],.]=>0 [[[.,[.,[.,[[.,.],.]]]],.],.]=>0 [[[.,[.,[[.,.],[.,.]]]],.],.]=>1 [[[.,[.,[[.,[.,.]],.]]],.],.]=>0 [[[.,[.,[[[.,.],.],.]]],.],.]=>0 [[[.,[[.,.],[.,[.,.]]]],.],.]=>1 [[[.,[[.,.],[[.,.],.]]],.],.]=>1 [[[.,[[.,[.,.]],[.,.]]],.],.]=>1 [[[.,[[[.,.],.],[.,.]]],.],.]=>1 [[[.,[[.,[.,[.,.]]],.]],.],.]=>0 [[[.,[[.,[[.,.],.]],.]],.],.]=>0 [[[.,[[[.,.],[.,.]],.]],.],.]=>1 [[[.,[[[.,[.,.]],.],.]],.],.]=>0 [[[.,[[[[.,.],.],.],.]],.],.]=>0 [[[[.,.],[.,[.,[.,.]]]],.],.]=>1 [[[[.,.],[.,[[.,.],.]]],.],.]=>1 [[[[.,.],[[.,.],[.,.]]],.],.]=>2 [[[[.,.],[[.,[.,.]],.]],.],.]=>1 [[[[.,.],[[[.,.],.],.]],.],.]=>1 [[[[.,[.,.]],[.,[.,.]]],.],.]=>1 [[[[.,[.,.]],[[.,.],.]],.],.]=>1 [[[[[.,.],.],[.,[.,.]]],.],.]=>1 [[[[[.,.],.],[[.,.],.]],.],.]=>1 [[[[.,[.,[.,.]]],[.,.]],.],.]=>1 [[[[.,[[.,.],.]],[.,.]],.],.]=>1 [[[[[.,.],[.,.]],[.,.]],.],.]=>2 [[[[[.,[.,.]],.],[.,.]],.],.]=>1 [[[[[[.,.],.],.],[.,.]],.],.]=>1 [[[[.,[.,[.,[.,.]]]],.],.],.]=>0 [[[[.,[.,[[.,.],.]]],.],.],.]=>0 [[[[.,[[.,.],[.,.]]],.],.],.]=>1 [[[[.,[[.,[.,.]],.]],.],.],.]=>0 [[[[.,[[[.,.],.],.]],.],.],.]=>0 [[[[[.,.],[.,[.,.]]],.],.],.]=>1 [[[[[.,.],[[.,.],.]],.],.],.]=>1 [[[[[.,[.,.]],[.,.]],.],.],.]=>1 [[[[[[.,.],.],[.,.]],.],.],.]=>1 [[[[[.,[.,[.,.]]],.],.],.],.]=>0 [[[[[.,[[.,.],.]],.],.],.],.]=>0 [[[[[[.,.],[.,.]],.],.],.],.]=>1 [[[[[[.,[.,.]],.],.],.],.],.]=>0 [[[[[[[.,.],.],.],.],.],.],.]=>0
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Description
The number of occurrences of the contiguous pattern [[.,.],[.,.]] in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Code
# BinaryTree -> Bool
def is_leaf(b):
    return b == BinaryTree('.')

# BinaryTree -> Int
def hat_count(b):
    if is_leaf(b):
        return 0
    if not is_leaf(b[0]) and not is_leaf(b[1]):
        return 1
    return 0

# (a -> a -> a) -> (BinaryTree -> a) -> BinaryTree -> a
def fold_tree(f, g, b):
    if is_leaf(b):
        return g(b)
    else:
        return f(g(b), f(fold_tree(f, g, b[0]), fold_tree(f, g, b[1])))

def statistic(b):
    return fold_tree(lambda a,b: a+b, hat_count, b)
Created
May 15, 2014 at 08:57 by Joseph Ching
Updated
May 30, 2016 at 11:58 by Christian Stump