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Statistic identifier: St000061

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Collection: Binary trees

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Description: The number of nodes on the left branch of a binary tree.

Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path. 

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References: [1]   Chapoton, F. Sur le nombre d'intervalles dans les treillis de Tamari [[MathSciNet:2264942]]

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Code:
def statistic(tree):
    if(not tree):
        return 0
    else: return 1 + statistic(tree[0])

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Statistic values:

[.,[.,.]]                 => 1
[[.,.],.]                 => 2
[.,[.,[.,.]]]             => 1
[.,[[.,.],.]]             => 1
[[.,.],[.,.]]             => 2
[[.,[.,.]],.]             => 2
[[[.,.],.],.]             => 3
[.,[.,[.,[.,.]]]]         => 1
[.,[.,[[.,.],.]]]         => 1
[.,[[.,.],[.,.]]]         => 1
[.,[[.,[.,.]],.]]         => 1
[.,[[[.,.],.],.]]         => 1
[[.,.],[.,[.,.]]]         => 2
[[.,.],[[.,.],.]]         => 2
[[.,[.,.]],[.,.]]         => 2
[[[.,.],.],[.,.]]         => 3
[[.,[.,[.,.]]],.]         => 2
[[.,[[.,.],.]],.]         => 2
[[[.,.],[.,.]],.]         => 3
[[[.,[.,.]],.],.]         => 3
[[[[.,.],.],.],.]         => 4
[.,[.,[.,[.,[.,.]]]]]     => 1
[.,[.,[.,[[.,.],.]]]]     => 1
[.,[.,[[.,.],[.,.]]]]     => 1
[.,[.,[[.,[.,.]],.]]]     => 1
[.,[.,[[[.,.],.],.]]]     => 1
[.,[[.,.],[.,[.,.]]]]     => 1
[.,[[.,.],[[.,.],.]]]     => 1
[.,[[.,[.,.]],[.,.]]]     => 1
[.,[[[.,.],.],[.,.]]]     => 1
[.,[[.,[.,[.,.]]],.]]     => 1
[.,[[.,[[.,.],.]],.]]     => 1
[.,[[[.,.],[.,.]],.]]     => 1
[.,[[[.,[.,.]],.],.]]     => 1
[.,[[[[.,.],.],.],.]]     => 1
[[.,.],[.,[.,[.,.]]]]     => 2
[[.,.],[.,[[.,.],.]]]     => 2
[[.,.],[[.,.],[.,.]]]     => 2
[[.,.],[[.,[.,.]],.]]     => 2
[[.,.],[[[.,.],.],.]]     => 2
[[.,[.,.]],[.,[.,.]]]     => 2
[[.,[.,.]],[[.,.],.]]     => 2
[[[.,.],.],[.,[.,.]]]     => 3
[[[.,.],.],[[.,.],.]]     => 3
[[.,[.,[.,.]]],[.,.]]     => 2
[[.,[[.,.],.]],[.,.]]     => 2
[[[.,.],[.,.]],[.,.]]     => 3
[[[.,[.,.]],.],[.,.]]     => 3
[[[[.,.],.],.],[.,.]]     => 4
[[.,[.,[.,[.,.]]]],.]     => 2
[[.,[.,[[.,.],.]]],.]     => 2
[[.,[[.,.],[.,.]]],.]     => 2
[[.,[[.,[.,.]],.]],.]     => 2
[[.,[[[.,.],.],.]],.]     => 2
[[[.,.],[.,[.,.]]],.]     => 3
[[[.,.],[[.,.],.]],.]     => 3
[[[.,[.,.]],[.,.]],.]     => 3
[[[[.,.],.],[.,.]],.]     => 4
[[[.,[.,[.,.]]],.],.]     => 3
[[[.,[[.,.],.]],.],.]     => 3
[[[[.,.],[.,.]],.],.]     => 4
[[[[.,[.,.]],.],.],.]     => 4
[[[[[.,.],.],.],.],.]     => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => 1
[.,[.,[.,[[[.,.],.],.]]]] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => 1
[.,[.,[[.,.],[[.,.],.]]]] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => 1
[.,[.,[[[.,.],.],[.,.]]]] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => 1
[.,[.,[[.,[[.,.],.]],.]]] => 1
[.,[.,[[[.,.],[.,.]],.]]] => 1
[.,[.,[[[.,[.,.]],.],.]]] => 1
[.,[.,[[[[.,.],.],.],.]]] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => 1
[.,[[.,.],[.,[[.,.],.]]]] => 1
[.,[[.,.],[[.,.],[.,.]]]] => 1
[.,[[.,.],[[.,[.,.]],.]]] => 1
[.,[[.,.],[[[.,.],.],.]]] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => 1
[.,[[.,[.,.]],[[.,.],.]]] => 1
[.,[[[.,.],.],[.,[.,.]]]] => 1
[.,[[[.,.],.],[[.,.],.]]] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => 1
[.,[[.,[[.,.],.]],[.,.]]] => 1
[.,[[[.,.],[.,.]],[.,.]]] => 1
[.,[[[.,[.,.]],.],[.,.]]] => 1
[.,[[[[.,.],.],.],[.,.]]] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => 1
[.,[[.,[.,[[.,.],.]]],.]] => 1
[.,[[.,[[.,.],[.,.]]],.]] => 1
[.,[[.,[[.,[.,.]],.]],.]] => 1
[.,[[.,[[[.,.],.],.]],.]] => 1
[.,[[[.,.],[.,[.,.]]],.]] => 1
[.,[[[.,.],[[.,.],.]],.]] => 1
[.,[[[.,[.,.]],[.,.]],.]] => 1
[.,[[[[.,.],.],[.,.]],.]] => 1
[.,[[[.,[.,[.,.]]],.],.]] => 1
[.,[[[.,[[.,.],.]],.],.]] => 1
[.,[[[[.,.],[.,.]],.],.]] => 1
[.,[[[[.,[.,.]],.],.],.]] => 1
[.,[[[[[.,.],.],.],.],.]] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => 2
[[.,.],[.,[.,[[.,.],.]]]] => 2
[[.,.],[.,[[.,.],[.,.]]]] => 2
[[.,.],[.,[[.,[.,.]],.]]] => 2
[[.,.],[.,[[[.,.],.],.]]] => 2
[[.,.],[[.,.],[.,[.,.]]]] => 2
[[.,.],[[.,.],[[.,.],.]]] => 2
[[.,.],[[.,[.,.]],[.,.]]] => 2
[[.,.],[[[.,.],.],[.,.]]] => 2
[[.,.],[[.,[.,[.,.]]],.]] => 2
[[.,.],[[.,[[.,.],.]],.]] => 2
[[.,.],[[[.,.],[.,.]],.]] => 2
[[.,.],[[[.,[.,.]],.],.]] => 2
[[.,.],[[[[.,.],.],.],.]] => 2
[[.,[.,.]],[.,[.,[.,.]]]] => 2
[[.,[.,.]],[.,[[.,.],.]]] => 2
[[.,[.,.]],[[.,.],[.,.]]] => 2
[[.,[.,.]],[[.,[.,.]],.]] => 2
[[.,[.,.]],[[[.,.],.],.]] => 2
[[[.,.],.],[.,[.,[.,.]]]] => 3
[[[.,.],.],[.,[[.,.],.]]] => 3
[[[.,.],.],[[.,.],[.,.]]] => 3
[[[.,.],.],[[.,[.,.]],.]] => 3
[[[.,.],.],[[[.,.],.],.]] => 3
[[.,[.,[.,.]]],[.,[.,.]]] => 2
[[.,[.,[.,.]]],[[.,.],.]] => 2
[[.,[[.,.],.]],[.,[.,.]]] => 2
[[.,[[.,.],.]],[[.,.],.]] => 2
[[[.,.],[.,.]],[.,[.,.]]] => 3
[[[.,.],[.,.]],[[.,.],.]] => 3
[[[.,[.,.]],.],[.,[.,.]]] => 3
[[[.,[.,.]],.],[[.,.],.]] => 3
[[[[.,.],.],.],[.,[.,.]]] => 4
[[[[.,.],.],.],[[.,.],.]] => 4
[[.,[.,[.,[.,.]]]],[.,.]] => 2
[[.,[.,[[.,.],.]]],[.,.]] => 2
[[.,[[.,.],[.,.]]],[.,.]] => 2
[[.,[[.,[.,.]],.]],[.,.]] => 2
[[.,[[[.,.],.],.]],[.,.]] => 2
[[[.,.],[.,[.,.]]],[.,.]] => 3
[[[.,.],[[.,.],.]],[.,.]] => 3
[[[.,[.,.]],[.,.]],[.,.]] => 3
[[[[.,.],.],[.,.]],[.,.]] => 4
[[[.,[.,[.,.]]],.],[.,.]] => 3
[[[.,[[.,.],.]],.],[.,.]] => 3
[[[[.,.],[.,.]],.],[.,.]] => 4
[[[[.,[.,.]],.],.],[.,.]] => 4
[[[[[.,.],.],.],.],[.,.]] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => 2
[[.,[.,[.,[[.,.],.]]]],.] => 2
[[.,[.,[[.,.],[.,.]]]],.] => 2
[[.,[.,[[.,[.,.]],.]]],.] => 2
[[.,[.,[[[.,.],.],.]]],.] => 2
[[.,[[.,.],[.,[.,.]]]],.] => 2
[[.,[[.,.],[[.,.],.]]],.] => 2
[[.,[[.,[.,.]],[.,.]]],.] => 2
[[.,[[[.,.],.],[.,.]]],.] => 2
[[.,[[.,[.,[.,.]]],.]],.] => 2
[[.,[[.,[[.,.],.]],.]],.] => 2
[[.,[[[.,.],[.,.]],.]],.] => 2
[[.,[[[.,[.,.]],.],.]],.] => 2
[[.,[[[[.,.],.],.],.]],.] => 2
[[[.,.],[.,[.,[.,.]]]],.] => 3
[[[.,.],[.,[[.,.],.]]],.] => 3
[[[.,.],[[.,.],[.,.]]],.] => 3
[[[.,.],[[.,[.,.]],.]],.] => 3
[[[.,.],[[[.,.],.],.]],.] => 3
[[[.,[.,.]],[.,[.,.]]],.] => 3
[[[.,[.,.]],[[.,.],.]],.] => 3
[[[[.,.],.],[.,[.,.]]],.] => 4
[[[[.,.],.],[[.,.],.]],.] => 4
[[[.,[.,[.,.]]],[.,.]],.] => 3
[[[.,[[.,.],.]],[.,.]],.] => 3
[[[[.,.],[.,.]],[.,.]],.] => 4
[[[[.,[.,.]],.],[.,.]],.] => 4
[[[[[.,.],.],.],[.,.]],.] => 5
[[[.,[.,[.,[.,.]]]],.],.] => 3
[[[.,[.,[[.,.],.]]],.],.] => 3
[[[.,[[.,.],[.,.]]],.],.] => 3
[[[.,[[.,[.,.]],.]],.],.] => 3
[[[.,[[[.,.],.],.]],.],.] => 3
[[[[.,.],[.,[.,.]]],.],.] => 4
[[[[.,.],[[.,.],.]],.],.] => 4
[[[[.,[.,.]],[.,.]],.],.] => 4
[[[[[.,.],.],[.,.]],.],.] => 5
[[[[.,[.,[.,.]]],.],.],.] => 4
[[[[.,[[.,.],.]],.],.],.] => 4
[[[[[.,.],[.,.]],.],.],.] => 5
[[[[[.,[.,.]],.],.],.],.] => 5
[[[[[[.,.],.],.],.],.],.] => 6

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Created: Apr 30, 2013 at 11:04 by Viviane Pons

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Last Updated: Oct 17, 2015 at 10:47 by Christian Stump