Identifier
- St000204: Binary trees ⟶ ℤ
Values
=>
Cc0010;cc-rep
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[[[[.,.],.],.],[[.,.],[.,.]]]=>1
[[[[.,.],.],.],[[.,[.,.]],.]]=>2
[[[[.,.],.],.],[[[.,.],.],.]]=>2
[[.,[.,[.,[.,.]]]],[.,[.,.]]]=>3
[[.,[.,[.,[.,.]]]],[[.,.],.]]=>4
[[.,[.,[[.,.],.]]],[.,[.,.]]]=>3
[[.,[.,[[.,.],.]]],[[.,.],.]]=>4
[[.,[[.,.],[.,.]]],[.,[.,.]]]=>3
[[.,[[.,.],[.,.]]],[[.,.],.]]=>4
[[.,[[.,[.,.]],.]],[.,[.,.]]]=>3
[[.,[[.,[.,.]],.]],[[.,.],.]]=>4
[[.,[[[.,.],.],.]],[.,[.,.]]]=>3
[[.,[[[.,.],.],.]],[[.,.],.]]=>4
[[[.,.],[.,[.,.]]],[.,[.,.]]]=>2
[[[.,.],[.,[.,.]]],[[.,.],.]]=>3
[[[.,.],[[.,.],.]],[.,[.,.]]]=>2
[[[.,.],[[.,.],.]],[[.,.],.]]=>3
[[[.,[.,.]],[.,.]],[.,[.,.]]]=>2
[[[.,[.,.]],[.,.]],[[.,.],.]]=>3
[[[[.,.],.],[.,.]],[.,[.,.]]]=>1
[[[[.,.],.],[.,.]],[[.,.],.]]=>2
[[[.,[.,[.,.]]],.],[.,[.,.]]]=>2
[[[.,[.,[.,.]]],.],[[.,.],.]]=>3
[[[.,[[.,.],.]],.],[.,[.,.]]]=>2
[[[.,[[.,.],.]],.],[[.,.],.]]=>3
[[[[.,.],[.,.]],.],[.,[.,.]]]=>1
[[[[.,.],[.,.]],.],[[.,.],.]]=>2
[[[[.,[.,.]],.],.],[.,[.,.]]]=>1
[[[[.,[.,.]],.],.],[[.,.],.]]=>2
[[[[[.,.],.],.],.],[.,[.,.]]]=>0
[[[[[.,.],.],.],.],[[.,.],.]]=>1
[[.,[.,[.,[.,[.,.]]]]],[.,.]]=>4
[[.,[.,[.,[[.,.],.]]]],[.,.]]=>4
[[.,[.,[[.,.],[.,.]]]],[.,.]]=>4
[[.,[.,[[.,[.,.]],.]]],[.,.]]=>4
[[.,[.,[[[.,.],.],.]]],[.,.]]=>4
[[.,[[.,.],[.,[.,.]]]],[.,.]]=>4
[[.,[[.,.],[[.,.],.]]],[.,.]]=>4
[[.,[[.,[.,.]],[.,.]]],[.,.]]=>4
[[.,[[[.,.],.],[.,.]]],[.,.]]=>4
[[.,[[.,[.,[.,.]]],.]],[.,.]]=>4
[[.,[[.,[[.,.],.]],.]],[.,.]]=>4
[[.,[[[.,.],[.,.]],.]],[.,.]]=>4
[[.,[[[.,[.,.]],.],.]],[.,.]]=>4
[[.,[[[[.,.],.],.],.]],[.,.]]=>4
[[[.,.],[.,[.,[.,.]]]],[.,.]]=>3
[[[.,.],[.,[[.,.],.]]],[.,.]]=>3
[[[.,.],[[.,.],[.,.]]],[.,.]]=>3
[[[.,.],[[.,[.,.]],.]],[.,.]]=>3
[[[.,.],[[[.,.],.],.]],[.,.]]=>3
[[[.,[.,.]],[.,[.,.]]],[.,.]]=>3
[[[.,[.,.]],[[.,.],.]],[.,.]]=>3
[[[[.,.],.],[.,[.,.]]],[.,.]]=>2
[[[[.,.],.],[[.,.],.]],[.,.]]=>2
[[[.,[.,[.,.]]],[.,.]],[.,.]]=>3
[[[.,[[.,.],.]],[.,.]],[.,.]]=>3
[[[[.,.],[.,.]],[.,.]],[.,.]]=>2
[[[[.,[.,.]],.],[.,.]],[.,.]]=>2
[[[[[.,.],.],.],[.,.]],[.,.]]=>1
[[[.,[.,[.,[.,.]]]],.],[.,.]]=>3
[[[.,[.,[[.,.],.]]],.],[.,.]]=>3
[[[.,[[.,.],[.,.]]],.],[.,.]]=>3
[[[.,[[.,[.,.]],.]],.],[.,.]]=>3
[[[.,[[[.,.],.],.]],.],[.,.]]=>3
[[[[.,.],[.,[.,.]]],.],[.,.]]=>2
[[[[.,.],[[.,.],.]],.],[.,.]]=>2
[[[[.,[.,.]],[.,.]],.],[.,.]]=>2
[[[[[.,.],.],[.,.]],.],[.,.]]=>1
[[[[.,[.,[.,.]]],.],.],[.,.]]=>2
[[[[.,[[.,.],.]],.],.],[.,.]]=>2
[[[[[.,.],[.,.]],.],.],[.,.]]=>1
[[[[[.,[.,.]],.],.],.],[.,.]]=>1
[[[[[[.,.],.],.],.],.],[.,.]]=>0
[[.,[.,[.,[.,[.,[.,.]]]]]],.]=>5
[[.,[.,[.,[.,[[.,.],.]]]]],.]=>5
[[.,[.,[.,[[.,.],[.,.]]]]],.]=>5
[[.,[.,[.,[[.,[.,.]],.]]]],.]=>5
[[.,[.,[.,[[[.,.],.],.]]]],.]=>5
[[.,[.,[[.,.],[.,[.,.]]]]],.]=>5
[[.,[.,[[.,.],[[.,.],.]]]],.]=>5
[[.,[.,[[.,[.,.]],[.,.]]]],.]=>5
[[.,[.,[[[.,.],.],[.,.]]]],.]=>5
[[.,[.,[[.,[.,[.,.]]],.]]],.]=>5
[[.,[.,[[.,[[.,.],.]],.]]],.]=>5
[[.,[.,[[[.,.],[.,.]],.]]],.]=>5
[[.,[.,[[[.,[.,.]],.],.]]],.]=>5
[[.,[.,[[[[.,.],.],.],.]]],.]=>5
[[.,[[.,.],[.,[.,[.,.]]]]],.]=>5
[[.,[[.,.],[.,[[.,.],.]]]],.]=>5
[[.,[[.,.],[[.,.],[.,.]]]],.]=>5
[[.,[[.,.],[[.,[.,.]],.]]],.]=>5
[[.,[[.,.],[[[.,.],.],.]]],.]=>5
[[.,[[.,[.,.]],[.,[.,.]]]],.]=>5
[[.,[[.,[.,.]],[[.,.],.]]],.]=>5
[[.,[[[.,.],.],[.,[.,.]]]],.]=>5
[[.,[[[.,.],.],[[.,.],.]]],.]=>5
[[.,[[.,[.,[.,.]]],[.,.]]],.]=>5
[[.,[[.,[[.,.],.]],[.,.]]],.]=>5
[[.,[[[.,.],[.,.]],[.,.]]],.]=>5
[[.,[[[.,[.,.]],.],[.,.]]],.]=>5
[[.,[[[[.,.],.],.],[.,.]]],.]=>5
[[.,[[.,[.,[.,[.,.]]]],.]],.]=>5
[[.,[[.,[.,[[.,.],.]]],.]],.]=>5
[[.,[[.,[[.,.],[.,.]]],.]],.]=>5
[[.,[[.,[[.,[.,.]],.]],.]],.]=>5
[[.,[[.,[[[.,.],.],.]],.]],.]=>5
[[.,[[[.,.],[.,[.,.]]],.]],.]=>5
[[.,[[[.,.],[[.,.],.]],.]],.]=>5
[[.,[[[.,[.,.]],[.,.]],.]],.]=>5
[[.,[[[[.,.],.],[.,.]],.]],.]=>5
[[.,[[[.,[.,[.,.]]],.],.]],.]=>5
[[.,[[[.,[[.,.],.]],.],.]],.]=>5
[[.,[[[[.,.],[.,.]],.],.]],.]=>5
[[.,[[[[.,[.,.]],.],.],.]],.]=>5
[[.,[[[[[.,.],.],.],.],.]],.]=>5
[[[.,.],[.,[.,[.,[.,.]]]]],.]=>4
[[[.,.],[.,[.,[[.,.],.]]]],.]=>4
[[[.,.],[.,[[.,.],[.,.]]]],.]=>4
[[[.,.],[.,[[.,[.,.]],.]]],.]=>4
[[[.,.],[.,[[[.,.],.],.]]],.]=>4
[[[.,.],[[.,.],[.,[.,.]]]],.]=>4
[[[.,.],[[.,.],[[.,.],.]]],.]=>4
[[[.,.],[[.,[.,.]],[.,.]]],.]=>4
[[[.,.],[[[.,.],.],[.,.]]],.]=>4
[[[.,.],[[.,[.,[.,.]]],.]],.]=>4
[[[.,.],[[.,[[.,.],.]],.]],.]=>4
[[[.,.],[[[.,.],[.,.]],.]],.]=>4
[[[.,.],[[[.,[.,.]],.],.]],.]=>4
[[[.,.],[[[[.,.],.],.],.]],.]=>4
[[[.,[.,.]],[.,[.,[.,.]]]],.]=>4
[[[.,[.,.]],[.,[[.,.],.]]],.]=>4
[[[.,[.,.]],[[.,.],[.,.]]],.]=>4
[[[.,[.,.]],[[.,[.,.]],.]],.]=>4
[[[.,[.,.]],[[[.,.],.],.]],.]=>4
[[[[.,.],.],[.,[.,[.,.]]]],.]=>3
[[[[.,.],.],[.,[[.,.],.]]],.]=>3
[[[[.,.],.],[[.,.],[.,.]]],.]=>3
[[[[.,.],.],[[.,[.,.]],.]],.]=>3
[[[[.,.],.],[[[.,.],.],.]],.]=>3
[[[.,[.,[.,.]]],[.,[.,.]]],.]=>4
[[[.,[.,[.,.]]],[[.,.],.]],.]=>4
[[[.,[[.,.],.]],[.,[.,.]]],.]=>4
[[[.,[[.,.],.]],[[.,.],.]],.]=>4
[[[[.,.],[.,.]],[.,[.,.]]],.]=>3
[[[[.,.],[.,.]],[[.,.],.]],.]=>3
[[[[.,[.,.]],.],[.,[.,.]]],.]=>3
[[[[.,[.,.]],.],[[.,.],.]],.]=>3
[[[[[.,.],.],.],[.,[.,.]]],.]=>2
[[[[[.,.],.],.],[[.,.],.]],.]=>2
[[[.,[.,[.,[.,.]]]],[.,.]],.]=>4
[[[.,[.,[[.,.],.]]],[.,.]],.]=>4
[[[.,[[.,.],[.,.]]],[.,.]],.]=>4
[[[.,[[.,[.,.]],.]],[.,.]],.]=>4
[[[.,[[[.,.],.],.]],[.,.]],.]=>4
[[[[.,.],[.,[.,.]]],[.,.]],.]=>3
[[[[.,.],[[.,.],.]],[.,.]],.]=>3
[[[[.,[.,.]],[.,.]],[.,.]],.]=>3
[[[[[.,.],.],[.,.]],[.,.]],.]=>2
[[[[.,[.,[.,.]]],.],[.,.]],.]=>3
[[[[.,[[.,.],.]],.],[.,.]],.]=>3
[[[[[.,.],[.,.]],.],[.,.]],.]=>2
[[[[[.,[.,.]],.],.],[.,.]],.]=>2
[[[[[[.,.],.],.],.],[.,.]],.]=>1
[[[.,[.,[.,[.,[.,.]]]]],.],.]=>4
[[[.,[.,[.,[[.,.],.]]]],.],.]=>4
[[[.,[.,[[.,.],[.,.]]]],.],.]=>4
[[[.,[.,[[.,[.,.]],.]]],.],.]=>4
[[[.,[.,[[[.,.],.],.]]],.],.]=>4
[[[.,[[.,.],[.,[.,.]]]],.],.]=>4
[[[.,[[.,.],[[.,.],.]]],.],.]=>4
[[[.,[[.,[.,.]],[.,.]]],.],.]=>4
[[[.,[[[.,.],.],[.,.]]],.],.]=>4
[[[.,[[.,[.,[.,.]]],.]],.],.]=>4
[[[.,[[.,[[.,.],.]],.]],.],.]=>4
[[[.,[[[.,.],[.,.]],.]],.],.]=>4
[[[.,[[[.,[.,.]],.],.]],.],.]=>4
[[[.,[[[[.,.],.],.],.]],.],.]=>4
[[[[.,.],[.,[.,[.,.]]]],.],.]=>3
[[[[.,.],[.,[[.,.],.]]],.],.]=>3
[[[[.,.],[[.,.],[.,.]]],.],.]=>3
[[[[.,.],[[.,[.,.]],.]],.],.]=>3
[[[[.,.],[[[.,.],.],.]],.],.]=>3
[[[[.,[.,.]],[.,[.,.]]],.],.]=>3
[[[[.,[.,.]],[[.,.],.]],.],.]=>3
[[[[[.,.],.],[.,[.,.]]],.],.]=>2
[[[[[.,.],.],[[.,.],.]],.],.]=>2
[[[[.,[.,[.,.]]],[.,.]],.],.]=>3
[[[[.,[[.,.],.]],[.,.]],.],.]=>3
[[[[[.,.],[.,.]],[.,.]],.],.]=>2
[[[[[.,[.,.]],.],[.,.]],.],.]=>2
[[[[[[.,.],.],.],[.,.]],.],.]=>1
[[[[.,[.,[.,[.,.]]]],.],.],.]=>3
[[[[.,[.,[[.,.],.]]],.],.],.]=>3
[[[[.,[[.,.],[.,.]]],.],.],.]=>3
[[[[.,[[.,[.,.]],.]],.],.],.]=>3
[[[[.,[[[.,.],.],.]],.],.],.]=>3
[[[[[.,.],[.,[.,.]]],.],.],.]=>2
[[[[[.,.],[[.,.],.]],.],.],.]=>2
[[[[[.,[.,.]],[.,.]],.],.],.]=>2
[[[[[[.,.],.],[.,.]],.],.],.]=>1
[[[[[.,[.,[.,.]]],.],.],.],.]=>2
[[[[[.,[[.,.],.]],.],.],.],.]=>2
[[[[[[.,.],[.,.]],.],.],.],.]=>1
[[[[[[.,[.,.]],.],.],.],.],.]=>1
[[[[[[[.,.],.],.],.],.],.],.]=>0
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Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus St000203The number of external nodes of a binary tree.. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
That is, the total number of nodes of the tree minus St000203The number of external nodes of a binary tree.. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
References
[1] Number of internal triangles in all triangulations of an (n+1)-gon. OEIS:A002058
Code
# BinaryTree -> Bool
def is_leaf(b):
return b == BinaryTree('.')
# BinaryTree -> Int
def get_external_node(b, flag):
count = 0;
if is_leaf(b):
return 0
else:
if (flag): # True counts the left side
if not is_leaf(b[0]):
return 1 + get_external_node(b[0], True)
else: # False counts the right side
if not is_leaf(b[1]):
return 1 + get_external_node(b[1], False)
return count
# BinaryTree -> Int
def external_node_count(b):
return 1 + get_external_node(b, True) + get_external_node(b, False)
def size(b):
if not b:
return 1
else:
return size(b[0])+size(b[1])
def statistic(b):
return size(b)-1-external_node_count(b)
Created
May 16, 2014 at 20:57 by Joseph Ching
Updated
Dec 30, 2016 at 10:54 by Christian Stump
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