edit this statistic or download as text // json
Identifier
Values
=>
Cc0010;cc-rep
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[.,[[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.],[.,.]]]=>72 [.,[[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]],[.,.]]]=>64 [.,[[[[.,.],[.,.]],[[[.,.],[.,.]],[.,.]]],[.,.]]]=>61 [.,[[[[.,[[.,.],[.,.]]],.],[[.,.],[.,.]]],[.,.]]]=>64 [.,[[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]],[.,.]]]=>61 [.,[[[[.,[[.,[[.,.],[.,.]]],.]],.],[.,.]],[.,.]]]=>73 [.,[[[[.,[[[.,.],[.,.]],[.,.]]],.],[.,.]],[.,.]]]=>70 [.,[[[[[.,.],[.,.]],[[.,.],[.,.]]],[.,.]],[.,.]]]=>63 [.,[[[[[.,[[.,.],[.,.]]],.],[.,.]],[.,.]],[.,.]]]=>68 [.,[[[[[[.,.],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]]=>65 [[.,[.,.]],[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]]=>71 [[.,[.,.]],[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]]=>68 [[.,[.,.]],[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]]=>61 [[.,[.,.]],[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]]=>66 [[.,[.,.]],[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.]]=>63 [[.,[.,.]],[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]]]=>55 [[.,[.,.]],[[[.,.],[.,.]],[[[.,.],[.,.]],[.,.]]]]=>52 [[.,[.,.]],[[[.,[[.,.],[.,.]]],.],[[.,.],[.,.]]]]=>55 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[.,[[.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]],.]]=>76 [.,[[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],.]]=>84 [.,[[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],.]]=>82 [.,[[.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]],.]]=>78 [.,[[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],.]]=>80 [.,[[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],.]]=>78 [.,[[[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]=>76 [.,[[[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]=>74 [.,[[[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]=>70 [.,[[[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]=>72 [.,[[[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]=>70 [.,[[[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]],.]]=>70 [.,[[[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]],.]]=>68 [.,[[[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]],.]]=>68 [.,[[[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]],.]]=>66 [.,[[[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]],.]]=>72 [.,[[[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]],.]]=>70 [.,[[[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]],.]]=>66 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[[[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]],[.,.]]=>58 [[[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]]=>56 [[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],[.,.]]=>72 [[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],[.,.]]=>70 [[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]],[.,.]]=>66 [[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],[.,.]]=>68 [[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],[.,.]]=>66 [[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]=>62 [[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]=>60 [[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]],[.,.]]=>60 [[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]=>58 [[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],[.,.]]=>66 [[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],[.,.]]=>64 [[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]=>60 [[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],[.,.]]=>62
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The external path length of a binary tree.
This is the sum of the lengths of all paths from the root to an external node, see Section 2.3.4.5 of [1].
This is also called the Sackin balance index of a rooted binary tree, see [2].
References
[1] Knuth, D. E. The art of computer programming. Vol. 1 MathSciNet:3077152
[2] Fischer, M. Extremal values of the Sackin balance index for rooted binary trees arXiv:1801.10418
Code
def statistic(T):
    return path_length_aux(T, 0) + 2*T.node_number()

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 16:02 by Martin Rubey
Updated
Mar 28, 2019 at 11:15 by Martin Rubey