edit this statistic or download as text // json
Identifier
Values
=>
Cc0010;cc-rep
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[.,[[[[.,[.,[.,.]]],[.,.]],.],.]]=>25 [.,[[[[.,[[.,.],.]],[.,.]],.],.]]=>25 [.,[[[[[.,.],[.,.]],[.,.]],.],.]]=>24 [.,[[[[[.,[.,.]],.],[.,.]],.],.]]=>25 [.,[[[[[[.,.],.],.],[.,.]],.],.]]=>25 [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]=>28 [.,[[[[.,[.,[[.,.],.]]],.],.],.]]=>28 [.,[[[[.,[[.,.],[.,.]]],.],.],.]]=>27 [.,[[[[.,[[.,[.,.]],.]],.],.],.]]=>28 [.,[[[[.,[[[.,.],.],.]],.],.],.]]=>28 [.,[[[[[.,.],[.,[.,.]]],.],.],.]]=>26 [.,[[[[[.,.],[[.,.],.]],.],.],.]]=>26 [.,[[[[[.,[.,.]],[.,.]],.],.],.]]=>26 [.,[[[[[[.,.],.],[.,.]],.],.],.]]=>26 [.,[[[[[.,[.,[.,.]]],.],.],.],.]]=>28 [.,[[[[[.,[[.,.],.]],.],.],.],.]]=>28 [.,[[[[[[.,.],[.,.]],.],.],.],.]]=>27 [.,[[[[[[.,[.,.]],.],.],.],.],.]]=>28 [.,[[[[[[[.,.],.],.],.],.],.],.]]=>28 [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>22 [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]=>22 [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]=>21 [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]=>22 [[.,.],[.,[.,[.,[[[.,.],.],.]]]]]=>22 [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]=>20 [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]=>20 [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]=>20 [[.,.],[.,[.,[[[.,.],.],[.,.]]]]]=>20 [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]=>22 [[.,.],[.,[.,[[.,[[.,.],.]],.]]]]=>22 [[.,.],[.,[.,[[[.,.],[.,.]],.]]]]=>21 [[.,.],[.,[.,[[[.,[.,.]],.],.]]]]=>22 [[.,.],[.,[.,[[[[.,.],.],.],.]]]]=>22 [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]=>19 [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]=>19 [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]=>18 [[.,.],[.,[[.,.],[[.,[.,.]],.]]]]=>19 [[.,.],[.,[[.,.],[[[.,.],.],.]]]]=>19 [[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]=>18 [[.,.],[.,[[.,[.,.]],[[.,.],.]]]]=>18 [[.,.],[.,[[[.,.],.],[.,[.,.]]]]]=>18 [[.,.],[.,[[[.,.],.],[[.,.],.]]]]=>18 [[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]=>19 [[.,.],[.,[[.,[[.,.],.]],[.,.]]]]=>19 [[.,.],[.,[[[.,.],[.,.]],[.,.]]]]=>18 [[.,.],[.,[[[.,[.,.]],.],[.,.]]]]=>19 [[.,.],[.,[[[[.,.],.],.],[.,.]]]]=>19 [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]=>22 [[.,.],[.,[[.,[.,[[.,.],.]]],.]]]=>22 [[.,.],[.,[[.,[[.,.],[.,.]]],.]]]=>21 [[.,.],[.,[[.,[[.,[.,.]],.]],.]]]=>22 [[.,.],[.,[[.,[[[.,.],.],.]],.]]]=>22 [[.,.],[.,[[[.,.],[.,[.,.]]],.]]]=>20 [[.,.],[.,[[[.,.],[[.,.],.]],.]]]=>20 [[.,.],[.,[[[.,[.,.]],[.,.]],.]]]=>20 [[.,.],[.,[[[[.,.],.],[.,.]],.]]]=>20 [[.,.],[.,[[[.,[.,[.,.]]],.],.]]]=>22 [[.,.],[.,[[[.,[[.,.],.]],.],.]]]=>22 [[.,.],[.,[[[[.,.],[.,.]],.],.]]]=>21 [[.,.],[.,[[[[.,[.,.]],.],.],.]]]=>22 [[.,.],[.,[[[[[.,.],.],.],.],.]]]=>22 [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]=>18 [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]=>18 [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]=>17 [[.,.],[[.,.],[.,[[.,[.,.]],.]]]]=>18 [[.,.],[[.,.],[.,[[[.,.],.],.]]]]=>18 [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]=>16 [[.,.],[[.,.],[[.,.],[[.,.],.]]]]=>16 [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]=>16 [[.,.],[[.,.],[[[.,.],.],[.,.]]]]=>16 [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]=>18 [[.,.],[[.,.],[[.,[[.,.],.]],.]]]=>18 [[.,.],[[.,.],[[[.,.],[.,.]],.]]]=>17 [[.,.],[[.,.],[[[.,[.,.]],.],.]]]=>18 [[.,.],[[.,.],[[[[.,.],.],.],.]]]=>18 [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]=>16 [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]=>16 [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]=>15 [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]=>16 [[.,.],[[.,[.,.]],[[[.,.],.],.]]]=>16 [[.,.],[[[.,.],.],[.,[.,[.,.]]]]]=>16 [[.,.],[[[.,.],.],[.,[[.,.],.]]]]=>16 [[.,.],[[[.,.],.],[[.,.],[.,.]]]]=>15 [[.,.],[[[.,.],.],[[.,[.,.]],.]]]=>16 [[.,.],[[[.,.],.],[[[.,.],.],.]]]=>16 [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]=>16 [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]=>16 [[.,.],[[.,[[.,.],.]],[.,[.,.]]]]=>16 [[.,.],[[.,[[.,.],.]],[[.,.],.]]]=>16 [[.,.],[[[.,.],[.,.]],[.,[.,.]]]]=>15 [[.,.],[[[.,.],[.,.]],[[.,.],.]]]=>15 [[.,.],[[[.,[.,.]],.],[.,[.,.]]]]=>16 [[.,.],[[[.,[.,.]],.],[[.,.],.]]]=>16 [[.,.],[[[[.,.],.],.],[.,[.,.]]]]=>16 [[.,.],[[[[.,.],.],.],[[.,.],.]]]=>16 [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]=>18 [[.,.],[[.,[.,[[.,.],.]]],[.,.]]]=>18 [[.,.],[[.,[[.,.],[.,.]]],[.,.]]]=>17 [[.,.],[[.,[[.,[.,.]],.]],[.,.]]]=>18 [[.,.],[[.,[[[.,.],.],.]],[.,.]]]=>18 [[.,.],[[[.,.],[.,[.,.]]],[.,.]]]=>16 [[.,.],[[[.,.],[[.,.],.]],[.,.]]]=>16 [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]=>16 [[.,.],[[[[.,.],.],[.,.]],[.,.]]]=>16 [[.,.],[[[.,[.,[.,.]]],.],[.,.]]]=>18 [[.,.],[[[.,[[.,.],.]],.],[.,.]]]=>18 [[.,.],[[[[.,.],[.,.]],.],[.,.]]]=>17 [[.,.],[[[[.,[.,.]],.],.],[.,.]]]=>18 [[.,.],[[[[[.,.],.],.],.],[.,.]]]=>18 [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]=>22 [[.,.],[[.,[.,[.,[[.,.],.]]]],.]]=>22 [[.,.],[[.,[.,[[.,.],[.,.]]]],.]]=>21 [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]=>22 [[.,.],[[.,[.,[[[.,.],.],.]]],.]]=>22 [[.,.],[[.,[[.,.],[.,[.,.]]]],.]]=>20 [[.,.],[[.,[[.,.],[[.,.],.]]],.]]=>20 [[.,.],[[.,[[.,[.,.]],[.,.]]],.]]=>20 [[.,.],[[.,[[[.,.],.],[.,.]]],.]]=>20 [[.,.],[[.,[[.,[.,[.,.]]],.]],.]]=>22 [[.,.],[[.,[[.,[[.,.],.]],.]],.]]=>22 [[.,.],[[.,[[[.,.],[.,.]],.]],.]]=>21 [[.,.],[[.,[[[.,[.,.]],.],.]],.]]=>22 [[.,.],[[.,[[[[.,.],.],.],.]],.]]=>22 [[.,.],[[[.,.],[.,[.,[.,.]]]],.]]=>19 [[.,.],[[[.,.],[.,[[.,.],.]]],.]]=>19 [[.,.],[[[.,.],[[.,.],[.,.]]],.]]=>18 [[.,.],[[[.,.],[[.,[.,.]],.]],.]]=>19 [[.,.],[[[.,.],[[[.,.],.],.]],.]]=>19 [[.,.],[[[.,[.,.]],[.,[.,.]]],.]]=>18 [[.,.],[[[.,[.,.]],[[.,.],.]],.]]=>18 [[.,.],[[[[.,.],.],[.,[.,.]]],.]]=>18 [[.,.],[[[[.,.],.],[[.,.],.]],.]]=>18 [[.,.],[[[.,[.,[.,.]]],[.,.]],.]]=>19 [[.,.],[[[.,[[.,.],.]],[.,.]],.]]=>19 [[.,.],[[[[.,.],[.,.]],[.,.]],.]]=>18 [[.,.],[[[[.,[.,.]],.],[.,.]],.]]=>19 [[.,.],[[[[[.,.],.],.],[.,.]],.]]=>19 [[.,.],[[[.,[.,[.,[.,.]]]],.],.]]=>22 [[.,.],[[[.,[.,[[.,.],.]]],.],.]]=>22 [[.,.],[[[.,[[.,.],[.,.]]],.],.]]=>21 [[.,.],[[[.,[[.,[.,.]],.]],.],.]]=>22 [[.,.],[[[.,[[[.,.],.],.]],.],.]]=>22 [[.,.],[[[[.,.],[.,[.,.]]],.],.]]=>20 [[.,.],[[[[.,.],[[.,.],.]],.],.]]=>20 [[.,.],[[[[.,[.,.]],[.,.]],.],.]]=>20 [[.,.],[[[[[.,.],.],[.,.]],.],.]]=>20 [[.,.],[[[[.,[.,[.,.]]],.],.],.]]=>22 [[.,.],[[[[.,[[.,.],.]],.],.],.]]=>22 [[.,.],[[[[[.,.],[.,.]],.],.],.]]=>21 [[.,.],[[[[[.,[.,.]],.],.],.],.]]=>22 [[.,.],[[[[[[.,.],.],.],.],.],.]]=>22 [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>18 [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]=>18 [[.,[.,.]],[.,[.,[[.,.],[.,.]]]]]=>17 [[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]=>18 [[.,[.,.]],[.,[.,[[[.,.],.],.]]]]=>18 [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]=>16 [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]=>16 [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]=>16 [[.,[.,.]],[.,[[[.,.],.],[.,.]]]]=>16 [[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]=>18 [[.,[.,.]],[.,[[.,[[.,.],.]],.]]]=>18 [[.,[.,.]],[.,[[[.,.],[.,.]],.]]]=>17 [[.,[.,.]],[.,[[[.,[.,.]],.],.]]]=>18 [[.,[.,.]],[.,[[[[.,.],.],.],.]]]=>18
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Description
The sum of the depths of the vertices (or total internal path length) of a binary tree.
The depth of a vertex is the number of edges to the tree's root, see Section 2.3.4.5 of [1] and [3].
This statistic is the very first entry of the OEIS, see [2]. Observe that there the term height is used instead.
References
[1] Knuth, D. E. The art of computer programming. Vol. 1 MathSciNet:3077152
[2] Normalized total height of all nodes in all rooted trees with n labeled nodes. OEIS:A000435
[3] wikipedia:Tree (data_structure)
Code
def statistic(T):
    return path_length_aux(T, 0)

def path_length_aux(T, n):
    return n+sum(path_length_aux(S, n+1) for S in T if not S.is_empty())
Created
Feb 22, 2016 at 15:54 by Martin Rubey
Updated
Aug 10, 2016 at 11:38 by Christian Stump