edit this statistic or download as text // json
Identifier
Values
=>
Cc0010;cc-rep
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[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]=>5 [[.,.],[.,[.,[[[[.,.],.],.],.]]]]=>5 [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]=>3 [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]=>3 [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]=>1 [[.,.],[.,[[.,.],[[.,[.,.]],.]]]]=>3 [[.,.],[.,[[.,.],[[[.,.],.],.]]]]=>3 [[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]=>3 [[.,.],[.,[[.,[.,.]],[[.,.],.]]]]=>3 [[.,.],[.,[[[.,.],.],[.,[.,.]]]]]=>3 [[.,.],[.,[[[.,.],.],[[.,.],.]]]]=>3 [[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]=>3 [[.,.],[.,[[.,[[.,.],.]],[.,.]]]]=>3 [[.,.],[.,[[[.,.],[.,.]],[.,.]]]]=>1 [[.,.],[.,[[[.,[.,.]],.],[.,.]]]]=>3 [[.,.],[.,[[[[.,.],.],.],[.,.]]]]=>3 [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]=>5 [[.,.],[.,[[.,[.,[[.,.],.]]],.]]]=>5 [[.,.],[.,[[.,[[.,.],[.,.]]],.]]]=>3 [[.,.],[.,[[.,[[.,[.,.]],.]],.]]]=>5 [[.,.],[.,[[.,[[[.,.],.],.]],.]]]=>5 [[.,.],[.,[[[.,.],[.,[.,.]]],.]]]=>3 [[.,.],[.,[[[.,.],[[.,.],.]],.]]]=>3 [[.,.],[.,[[[.,[.,.]],[.,.]],.]]]=>3 [[.,.],[.,[[[[.,.],.],[.,.]],.]]]=>3 [[.,.],[.,[[[.,[.,[.,.]]],.],.]]]=>5 [[.,.],[.,[[[.,[[.,.],.]],.],.]]]=>5 [[.,.],[.,[[[[.,.],[.,.]],.],.]]]=>3 [[.,.],[.,[[[[.,[.,.]],.],.],.]]]=>5 [[.,.],[.,[[[[[.,.],.],.],.],.]]]=>5 [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]=>3 [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]=>3 [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]=>1 [[.,.],[[.,.],[.,[[.,[.,.]],.]]]]=>3 [[.,.],[[.,.],[.,[[[.,.],.],.]]]]=>3 [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]=>1 [[.,.],[[.,.],[[.,.],[[.,.],.]]]]=>1 [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]=>1 [[.,.],[[.,.],[[[.,.],.],[.,.]]]]=>1 [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]=>3 [[.,.],[[.,.],[[.,[[.,.],.]],.]]]=>3 [[.,.],[[.,.],[[[.,.],[.,.]],.]]]=>1 [[.,.],[[.,.],[[[.,[.,.]],.],.]]]=>3 [[.,.],[[.,.],[[[[.,.],.],.],.]]]=>3 [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]=>3 [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]=>3 [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]=>1 [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]=>3 [[.,.],[[.,[.,.]],[[[.,.],.],.]]]=>3 [[.,.],[[[.,.],.],[.,[.,[.,.]]]]]=>3 [[.,.],[[[.,.],.],[.,[[.,.],.]]]]=>3 [[.,.],[[[.,.],.],[[.,.],[.,.]]]]=>1 [[.,.],[[[.,.],.],[[.,[.,.]],.]]]=>3 [[.,.],[[[.,.],.],[[[.,.],.],.]]]=>3 [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]=>3 [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]=>3 [[.,.],[[.,[[.,.],.]],[.,[.,.]]]]=>3 [[.,.],[[.,[[.,.],.]],[[.,.],.]]]=>3 [[.,.],[[[.,.],[.,.]],[.,[.,.]]]]=>1 [[.,.],[[[.,.],[.,.]],[[.,.],.]]]=>1 [[.,.],[[[.,[.,.]],.],[.,[.,.]]]]=>3 [[.,.],[[[.,[.,.]],.],[[.,.],.]]]=>3 [[.,.],[[[[.,.],.],.],[.,[.,.]]]]=>3 [[.,.],[[[[.,.],.],.],[[.,.],.]]]=>3 [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]=>3 [[.,.],[[.,[.,[[.,.],.]]],[.,.]]]=>3 [[.,.],[[.,[[.,.],[.,.]]],[.,.]]]=>1 [[.,.],[[.,[[.,[.,.]],.]],[.,.]]]=>3 [[.,.],[[.,[[[.,.],.],.]],[.,.]]]=>3 [[.,.],[[[.,.],[.,[.,.]]],[.,.]]]=>1 [[.,.],[[[.,.],[[.,.],.]],[.,.]]]=>1 [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]=>1 [[.,.],[[[[.,.],.],[.,.]],[.,.]]]=>1 [[.,.],[[[.,[.,[.,.]]],.],[.,.]]]=>3 [[.,.],[[[.,[[.,.],.]],.],[.,.]]]=>3 [[.,.],[[[[.,.],[.,.]],.],[.,.]]]=>1 [[.,.],[[[[.,[.,.]],.],.],[.,.]]]=>3 [[.,.],[[[[[.,.],.],.],.],[.,.]]]=>3 [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]=>5 [[.,.],[[.,[.,[.,[[.,.],.]]]],.]]=>5 [[.,.],[[.,[.,[[.,.],[.,.]]]],.]]=>3 [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]=>5 [[.,.],[[.,[.,[[[.,.],.],.]]],.]]=>5 [[.,.],[[.,[[.,.],[.,[.,.]]]],.]]=>3 [[.,.],[[.,[[.,.],[[.,.],.]]],.]]=>3 [[.,.],[[.,[[.,[.,.]],[.,.]]],.]]=>3 [[.,.],[[.,[[[.,.],.],[.,.]]],.]]=>3 [[.,.],[[.,[[.,[.,[.,.]]],.]],.]]=>5 [[.,.],[[.,[[.,[[.,.],.]],.]],.]]=>5 [[.,.],[[.,[[[.,.],[.,.]],.]],.]]=>3 [[.,.],[[.,[[[.,[.,.]],.],.]],.]]=>5 [[.,.],[[.,[[[[.,.],.],.],.]],.]]=>5 [[.,.],[[[.,.],[.,[.,[.,.]]]],.]]=>3 [[.,.],[[[.,.],[.,[[.,.],.]]],.]]=>3 [[.,.],[[[.,.],[[.,.],[.,.]]],.]]=>1 [[.,.],[[[.,.],[[.,[.,.]],.]],.]]=>3 [[.,.],[[[.,.],[[[.,.],.],.]],.]]=>3 [[.,.],[[[.,[.,.]],[.,[.,.]]],.]]=>3 [[.,.],[[[.,[.,.]],[[.,.],.]],.]]=>3 [[.,.],[[[[.,.],.],[.,[.,.]]],.]]=>3 [[.,.],[[[[.,.],.],[[.,.],.]],.]]=>3 [[.,.],[[[.,[.,[.,.]]],[.,.]],.]]=>3 [[.,.],[[[.,[[.,.],.]],[.,.]],.]]=>3 [[.,.],[[[[.,.],[.,.]],[.,.]],.]]=>1 [[.,.],[[[[.,[.,.]],.],[.,.]],.]]=>3 [[.,.],[[[[[.,.],.],.],[.,.]],.]]=>3 [[.,.],[[[.,[.,[.,[.,.]]]],.],.]]=>5 [[.,.],[[[.,[.,[[.,.],.]]],.],.]]=>5 [[.,.],[[[.,[[.,.],[.,.]]],.],.]]=>3 [[.,.],[[[.,[[.,[.,.]],.]],.],.]]=>5 [[.,.],[[[.,[[[.,.],.],.]],.],.]]=>5 [[.,.],[[[[.,.],[.,[.,.]]],.],.]]=>3 [[.,.],[[[[.,.],[[.,.],.]],.],.]]=>3 [[.,.],[[[[.,[.,.]],[.,.]],.],.]]=>3 [[.,.],[[[[[.,.],.],[.,.]],.],.]]=>3 [[.,.],[[[[.,[.,[.,.]]],.],.],.]]=>5 [[.,.],[[[[.,[[.,.],.]],.],.],.]]=>5 [[.,.],[[[[[.,.],[.,.]],.],.],.]]=>3 [[.,.],[[[[[.,[.,.]],.],.],.],.]]=>5 [[.,.],[[[[[[.,.],.],.],.],.],.]]=>5 [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>5 [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]=>5 [[.,[.,.]],[.,[.,[[.,.],[.,.]]]]]=>3 [[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]=>5 [[.,[.,.]],[.,[.,[[[.,.],.],.]]]]=>5 [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]=>3 [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]=>3 [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]=>3 [[.,[.,.]],[.,[[[.,.],.],[.,.]]]]=>3 [[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]=>5 [[.,[.,.]],[.,[[.,[[.,.],.]],.]]]=>5 [[.,[.,.]],[.,[[[.,.],[.,.]],.]]]=>3 [[.,[.,.]],[.,[[[.,[.,.]],.],.]]]=>5 [[.,[.,.]],[.,[[[[.,.],.],.],.]]]=>5 [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]=>3
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Description
The number of vertices with out-degree 1 in a binary tree.
See the references for several connections of this statistic.
In particular, the number $T(n,k)$ of binary trees with $n$ vertices and $k$ out-degree $1$ vertices is given by $T(n,k) = 0$ for $n-k$ odd and
$$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$$
for $n-k$ is even.
References
[1] Triangle read by rows: T(n,k) is the number of binary trees with n edges and having k vertices of outdegree 1 (n>=0, k>=0). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child. OEIS:A121448
[2] Copeland, T. Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)? MathOverflow:230714
Code
def statistic(tree):
    return tree.graph(with_leaves=False).out_degree_sequence().count(1)
Created
Feb 10, 2016 at 09:37 by Christian Stump
Updated
Feb 10, 2016 at 09:57 by Christian Stump