***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000050 ----------------------------------------------------------------------------- Collection: Binary trees ----------------------------------------------------------------------------- Description: The depth or height of a binary tree. The depth (or height) of a binary tree is the maximal depth (or height) of one of its vertices. The '''height''' of a vertex is the number of edges on the longest path between that node and a leaf. The '''depth''' of a vertex is the number of edges from the vertex to the root. See [1] and [2] for this terminology. The depth (or height) of a tree $T$ can be recursively defined: $\operatorname{depth}(T) = 0$ if $T$ is empty and $$\operatorname{depth}(T) = 1 + max(\operatorname{depth}(L),\operatorname{depth}(R))$$ if $T$ is nonempty with left and right subtrees $L$ and $R$, respectively. The upper and lower bounds on the depth of a binary tree $T$ of size $n$ are $log_2(n) \leq \operatorname{depth}(T) \leq n$. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Tree (data_structure)]] [2] [[http://stackoverflow.com/questions/2603692|Depth and height of tree]] ----------------------------------------------------------------------------- Code: def statistic(t): return t.depth() ----------------------------------------------------------------------------- Statistic values: [.,.] => 1 [.,[.,.]] => 2 [[.,.],.] => 2 [.,[.,[.,.]]] => 3 [.,[[.,.],.]] => 3 [[.,.],[.,.]] => 2 [[.,[.,.]],.] => 3 [[[.,.],.],.] => 3 [.,[.,[.,[.,.]]]] => 4 [.,[.,[[.,.],.]]] => 4 [.,[[.,.],[.,.]]] => 3 [.,[[.,[.,.]],.]] => 4 [.,[[[.,.],.],.]] => 4 [[.,.],[.,[.,.]]] => 3 [[.,.],[[.,.],.]] => 3 [[.,[.,.]],[.,.]] => 3 [[[.,.],.],[.,.]] => 3 [[.,[.,[.,.]]],.] => 4 [[.,[[.,.],.]],.] => 4 [[[.,.],[.,.]],.] => 3 [[[.,[.,.]],.],.] => 4 [[[[.,.],.],.],.] => 4 [.,[.,[.,[.,[.,.]]]]] => 5 [.,[.,[.,[[.,.],.]]]] => 5 [.,[.,[[.,.],[.,.]]]] => 4 [.,[.,[[.,[.,.]],.]]] => 5 [.,[.,[[[.,.],.],.]]] => 5 [.,[[.,.],[.,[.,.]]]] => 4 [.,[[.,.],[[.,.],.]]] => 4 [.,[[.,[.,.]],[.,.]]] => 4 [.,[[[.,.],.],[.,.]]] => 4 [.,[[.,[.,[.,.]]],.]] => 5 [.,[[.,[[.,.],.]],.]] => 5 [.,[[[.,.],[.,.]],.]] => 4 [.,[[[.,[.,.]],.],.]] => 5 [.,[[[[.,.],.],.],.]] => 5 [[.,.],[.,[.,[.,.]]]] => 4 [[.,.],[.,[[.,.],.]]] => 4 [[.,.],[[.,.],[.,.]]] => 3 [[.,.],[[.,[.,.]],.]] => 4 [[.,.],[[[.,.],.],.]] => 4 [[.,[.,.]],[.,[.,.]]] => 3 [[.,[.,.]],[[.,.],.]] => 3 [[[.,.],.],[.,[.,.]]] => 3 [[[.,.],.],[[.,.],.]] => 3 [[.,[.,[.,.]]],[.,.]] => 4 [[.,[[.,.],.]],[.,.]] => 4 [[[.,.],[.,.]],[.,.]] => 3 [[[.,[.,.]],.],[.,.]] => 4 [[[[.,.],.],.],[.,.]] => 4 [[.,[.,[.,[.,.]]]],.] => 5 [[.,[.,[[.,.],.]]],.] => 5 [[.,[[.,.],[.,.]]],.] => 4 [[.,[[.,[.,.]],.]],.] => 5 [[.,[[[.,.],.],.]],.] => 5 [[[.,.],[.,[.,.]]],.] => 4 [[[.,.],[[.,.],.]],.] => 4 [[[.,[.,.]],[.,.]],.] => 4 [[[[.,.],.],[.,.]],.] => 4 [[[.,[.,[.,.]]],.],.] => 5 [[[.,[[.,.],.]],.],.] => 5 [[[[.,.],[.,.]],.],.] => 4 [[[[.,[.,.]],.],.],.] => 5 [[[[[.,.],.],.],.],.] => 5 [.,[.,[.,[.,[.,[.,.]]]]]] => 6 [.,[.,[.,[.,[[.,.],.]]]]] => 6 [.,[.,[.,[[.,.],[.,.]]]]] => 5 [.,[.,[.,[[.,[.,.]],.]]]] => 6 [.,[.,[.,[[[.,.],.],.]]]] => 6 [.,[.,[[.,.],[.,[.,.]]]]] => 5 [.,[.,[[.,.],[[.,.],.]]]] => 5 [.,[.,[[.,[.,.]],[.,.]]]] => 5 [.,[.,[[[.,.],.],[.,.]]]] => 5 [.,[.,[[.,[.,[.,.]]],.]]] => 6 [.,[.,[[.,[[.,.],.]],.]]] => 6 [.,[.,[[[.,.],[.,.]],.]]] => 5 [.,[.,[[[.,[.,.]],.],.]]] => 6 [.,[.,[[[[.,.],.],.],.]]] => 6 [.,[[.,.],[.,[.,[.,.]]]]] => 5 [.,[[.,.],[.,[[.,.],.]]]] => 5 [.,[[.,.],[[.,.],[.,.]]]] => 4 [.,[[.,.],[[.,[.,.]],.]]] => 5 [.,[[.,.],[[[.,.],.],.]]] => 5 [.,[[.,[.,.]],[.,[.,.]]]] => 4 [.,[[.,[.,.]],[[.,.],.]]] => 4 [.,[[[.,.],.],[.,[.,.]]]] => 4 [.,[[[.,.],.],[[.,.],.]]] => 4 [.,[[.,[.,[.,.]]],[.,.]]] => 5 [.,[[.,[[.,.],.]],[.,.]]] => 5 [.,[[[.,.],[.,.]],[.,.]]] => 4 [.,[[[.,[.,.]],.],[.,.]]] => 5 [.,[[[[.,.],.],.],[.,.]]] => 5 [.,[[.,[.,[.,[.,.]]]],.]] => 6 [.,[[.,[.,[[.,.],.]]],.]] => 6 [.,[[.,[[.,.],[.,.]]],.]] => 5 [.,[[.,[[.,[.,.]],.]],.]] => 6 [.,[[.,[[[.,.],.],.]],.]] => 6 [.,[[[.,.],[.,[.,.]]],.]] => 5 [.,[[[.,.],[[.,.],.]],.]] => 5 [.,[[[.,[.,.]],[.,.]],.]] => 5 [.,[[[[.,.],.],[.,.]],.]] => 5 [.,[[[.,[.,[.,.]]],.],.]] => 6 [.,[[[.,[[.,.],.]],.],.]] => 6 [.,[[[[.,.],[.,.]],.],.]] => 5 [.,[[[[.,[.,.]],.],.],.]] => 6 [.,[[[[[.,.],.],.],.],.]] => 6 [[.,.],[.,[.,[.,[.,.]]]]] => 5 [[.,.],[.,[.,[[.,.],.]]]] => 5 [[.,.],[.,[[.,.],[.,.]]]] => 4 [[.,.],[.,[[.,[.,.]],.]]] => 5 [[.,.],[.,[[[.,.],.],.]]] => 5 [[.,.],[[.,.],[.,[.,.]]]] => 4 [[.,.],[[.,.],[[.,.],.]]] => 4 [[.,.],[[.,[.,.]],[.,.]]] => 4 [[.,.],[[[.,.],.],[.,.]]] => 4 [[.,.],[[.,[.,[.,.]]],.]] => 5 [[.,.],[[.,[[.,.],.]],.]] => 5 [[.,.],[[[.,.],[.,.]],.]] => 4 [[.,.],[[[.,[.,.]],.],.]] => 5 [[.,.],[[[[.,.],.],.],.]] => 5 [[.,[.,.]],[.,[.,[.,.]]]] => 4 [[.,[.,.]],[.,[[.,.],.]]] => 4 [[.,[.,.]],[[.,.],[.,.]]] => 3 [[.,[.,.]],[[.,[.,.]],.]] => 4 [[.,[.,.]],[[[.,.],.],.]] => 4 [[[.,.],.],[.,[.,[.,.]]]] => 4 [[[.,.],.],[.,[[.,.],.]]] => 4 [[[.,.],.],[[.,.],[.,.]]] => 3 [[[.,.],.],[[.,[.,.]],.]] => 4 [[[.,.],.],[[[.,.],.],.]] => 4 [[.,[.,[.,.]]],[.,[.,.]]] => 4 [[.,[.,[.,.]]],[[.,.],.]] => 4 [[.,[[.,.],.]],[.,[.,.]]] => 4 [[.,[[.,.],.]],[[.,.],.]] => 4 [[[.,.],[.,.]],[.,[.,.]]] => 3 [[[.,.],[.,.]],[[.,.],.]] => 3 [[[.,[.,.]],.],[.,[.,.]]] => 4 [[[.,[.,.]],.],[[.,.],.]] => 4 [[[[.,.],.],.],[.,[.,.]]] => 4 [[[[.,.],.],.],[[.,.],.]] => 4 [[.,[.,[.,[.,.]]]],[.,.]] => 5 [[.,[.,[[.,.],.]]],[.,.]] => 5 [[.,[[.,.],[.,.]]],[.,.]] => 4 [[.,[[.,[.,.]],.]],[.,.]] => 5 [[.,[[[.,.],.],.]],[.,.]] => 5 [[[.,.],[.,[.,.]]],[.,.]] => 4 [[[.,.],[[.,.],.]],[.,.]] => 4 [[[.,[.,.]],[.,.]],[.,.]] => 4 [[[[.,.],.],[.,.]],[.,.]] => 4 [[[.,[.,[.,.]]],.],[.,.]] => 5 [[[.,[[.,.],.]],.],[.,.]] => 5 [[[[.,.],[.,.]],.],[.,.]] => 4 [[[[.,[.,.]],.],.],[.,.]] => 5 [[[[[.,.],.],.],.],[.,.]] => 5 [[.,[.,[.,[.,[.,.]]]]],.] => 6 [[.,[.,[.,[[.,.],.]]]],.] => 6 [[.,[.,[[.,.],[.,.]]]],.] => 5 [[.,[.,[[.,[.,.]],.]]],.] => 6 [[.,[.,[[[.,.],.],.]]],.] => 6 [[.,[[.,.],[.,[.,.]]]],.] => 5 [[.,[[.,.],[[.,.],.]]],.] => 5 [[.,[[.,[.,.]],[.,.]]],.] => 5 [[.,[[[.,.],.],[.,.]]],.] => 5 [[.,[[.,[.,[.,.]]],.]],.] => 6 [[.,[[.,[[.,.],.]],.]],.] => 6 [[.,[[[.,.],[.,.]],.]],.] => 5 [[.,[[[.,[.,.]],.],.]],.] => 6 [[.,[[[[.,.],.],.],.]],.] => 6 [[[.,.],[.,[.,[.,.]]]],.] => 5 [[[.,.],[.,[[.,.],.]]],.] => 5 [[[.,.],[[.,.],[.,.]]],.] => 4 [[[.,.],[[.,[.,.]],.]],.] => 5 [[[.,.],[[[.,.],.],.]],.] => 5 [[[.,[.,.]],[.,[.,.]]],.] => 4 [[[.,[.,.]],[[.,.],.]],.] => 4 [[[[.,.],.],[.,[.,.]]],.] => 4 [[[[.,.],.],[[.,.],.]],.] => 4 [[[.,[.,[.,.]]],[.,.]],.] => 5 [[[.,[[.,.],.]],[.,.]],.] => 5 [[[[.,.],[.,.]],[.,.]],.] => 4 [[[[.,[.,.]],.],[.,.]],.] => 5 [[[[[.,.],.],.],[.,.]],.] => 5 [[[.,[.,[.,[.,.]]]],.],.] => 6 [[[.,[.,[[.,.],.]]],.],.] => 6 [[[.,[[.,.],[.,.]]],.],.] => 5 [[[.,[[.,[.,.]],.]],.],.] => 6 [[[.,[[[.,.],.],.]],.],.] => 6 [[[[.,.],[.,[.,.]]],.],.] => 5 [[[[.,.],[[.,.],.]],.],.] => 5 [[[[.,[.,.]],[.,.]],.],.] => 5 [[[[[.,.],.],[.,.]],.],.] => 5 [[[[.,[.,[.,.]]],.],.],.] => 6 [[[[.,[[.,.],.]],.],.],.] => 6 [[[[[.,.],[.,.]],.],.],.] => 5 [[[[[.,[.,.]],.],.],.],.] => 6 [[[[[[.,.],.],.],.],.],.] => 6 [.,[[[.,.],[.,.]],[.,[.,.]]]] => 4 [.,[[[.,.],[.,.]],[[.,.],.]]] => 4 [.,[[.,[[.,.],[.,.]]],[.,.]]] => 5 [.,[[.,[[[.,.],.],.]],[.,.]]] => 6 [.,[[[.,.],[[.,.],.]],[.,.]]] => 5 [.,[[[.,[.,.]],[.,.]],[.,.]]] => 5 [.,[[[[.,.],.],[.,.]],[.,.]]] => 5 [.,[[[.,[[.,.],.]],.],[.,.]]] => 6 [.,[[[[.,.],[.,.]],.],[.,.]]] => 5 [.,[[[[.,[.,.]],.],.],[.,.]]] => 6 [.,[[[[[.,.],.],.],.],[.,.]]] => 6 [.,[[.,[[[[.,.],.],.],.]],.]] => 7 [.,[[[[.,.],[.,.]],[.,.]],.]] => 5 [.,[[[[[.,.],.],.],[.,.]],.]] => 6 [.,[[[.,[[.,[.,.]],.]],.],.]] => 7 [.,[[[.,[[[.,.],.],.]],.],.]] => 7 [.,[[[[.,.],[[.,.],.]],.],.]] => 6 [.,[[[[.,[.,.]],[.,.]],.],.]] => 6 [.,[[[[[.,.],.],[.,.]],.],.]] => 6 [.,[[[[.,[.,[.,.]]],.],.],.]] => 7 [.,[[[[.,[[.,.],.]],.],.],.]] => 7 [.,[[[[[.,.],[.,.]],.],.],.]] => 6 [.,[[[[[.,[.,.]],.],.],.],.]] => 7 [.,[[[[[[.,.],.],.],.],.],.]] => 7 [[.,.],[.,[.,[.,[.,[.,.]]]]]] => 6 [[.,.],[.,[.,[.,[[.,.],.]]]]] => 6 [[.,.],[.,[.,[[.,.],[.,.]]]]] => 5 [[.,.],[.,[.,[[.,[.,.]],.]]]] => 6 [[.,.],[.,[.,[[[.,.],.],.]]]] => 6 [[.,.],[.,[[.,.],[.,[.,.]]]]] => 5 [[.,.],[.,[[.,.],[[.,.],.]]]] => 5 [[.,.],[.,[[.,[.,.]],[.,.]]]] => 5 [[.,.],[.,[[.,[.,[.,.]]],.]]] => 6 [[.,.],[.,[[.,[[.,.],.]],.]]] => 6 [[.,.],[[.,.],[.,[.,[.,.]]]]] => 5 [[.,.],[[.,[.,[.,[.,.]]]],.]] => 6 [[.,.],[[[[[.,.],.],.],.],.]] => 6 [[.,[.,.]],[.,[.,[.,[.,.]]]]] => 5 [[.,[[[.,.],.],.]],[.,[.,.]]] => 5 [[[.,.],[.,[.,.]]],[.,[.,.]]] => 4 [[[.,.],[.,[.,.]]],[[.,.],.]] => 4 [[[.,[[.,.],.]],.],[.,[.,.]]] => 5 [[[[.,[.,.]],.],.],[.,[.,.]]] => 5 [[[[[.,.],.],.],.],[.,[.,.]]] => 5 [[[[[.,.],.],.],.],[[.,.],.]] => 5 [[.,[.,[.,[.,[.,.]]]]],[.,.]] => 6 [[.,[[.,.],[.,[.,.]]]],[.,.]] => 5 [[.,[[[[.,.],.],.],.]],[.,.]] => 6 [[[.,.],[.,[[.,.],.]]],[.,.]] => 5 [[[.,.],[[.,[.,.]],.]],[.,.]] => 5 [[[.,[.,.]],[.,[.,.]]],[.,.]] => 4 [[[[.,.],.],[.,[.,.]]],[.,.]] => 4 [[[[[.,.],.],.],[.,.]],[.,.]] => 5 [[[.,[[.,[.,.]],.]],.],[.,.]] => 6 [[[.,[[[.,.],.],.]],.],[.,.]] => 6 [[[[.,.],[.,[.,.]]],.],[.,.]] => 5 [[[[.,.],[[.,.],.]],.],[.,.]] => 5 [[[[.,[.,.]],[.,.]],.],[.,.]] => 5 [[[[[.,.],.],[.,.]],.],[.,.]] => 5 [[[[.,[.,[.,.]]],.],.],[.,.]] => 6 [[[[.,[[.,.],.]],.],.],[.,.]] => 6 [[[[[.,.],[.,.]],.],.],[.,.]] => 5 [[[[[.,[.,.]],.],.],.],[.,.]] => 6 [[[[[[.,.],.],.],.],.],[.,.]] => 6 [[.,[.,[.,[.,[.,[.,.]]]]]],.] => 7 [[.,[.,[.,[.,[[.,.],.]]]]],.] => 7 [[.,[.,[.,[[.,.],[.,.]]]]],.] => 6 [[.,[.,[.,[[.,[.,.]],.]]]],.] => 7 [[.,[.,[.,[[[.,.],.],.]]]],.] => 7 [[.,[.,[[.,.],[.,[.,.]]]]],.] => 6 [[.,[.,[[.,.],[[.,.],.]]]],.] => 6 [[.,[.,[[.,[.,.]],[.,.]]]],.] => 6 [[.,[.,[[.,[.,[.,.]]],.]]],.] => 7 [[.,[.,[[.,[[.,.],.]],.]]],.] => 7 [[.,[[.,.],[.,[.,[.,.]]]]],.] => 6 [[.,[[[.,.],[.,.]],[.,.]]],.] => 5 [[.,[[.,[.,[.,[.,.]]]],.]],.] => 7 [[[[.,.],[.,[.,.]]],[.,.]],.] => 5 [.,[[[.,.],[.,.]],[.,[.,[.,.]]]]] => 5 [.,[[[.,.],[.,.]],[.,[[.,.],.]]]] => 5 [.,[[[.,.],[.,.]],[[.,.],[.,.]]]] => 4 [.,[[[.,.],[.,.]],[[.,[.,.]],.]]] => 5 [.,[[[.,.],[.,.]],[[[.,.],.],.]]] => 5 [.,[[[[.,.],.],[.,.]],[[.,.],.]]] => 5 [.,[[[[.,.],[.,.]],.],[[.,.],.]]] => 5 [.,[[[[.,.],[.,.]],[.,.]],[.,.]]] => 5 [.,[[[.,[[.,.],[.,.]]],.],[.,.]]] => 6 [.,[[.,[[[.,.],[.,.]],[.,.]]],.]] => 6 [.,[[.,[[.,[[.,.],[.,.]]],.]],.]] => 7 [.,[[.,[[.,[[.,[.,.]],.]],.]],.]] => 8 [.,[[.,[[[.,[.,.]],[.,.]],.]],.]] => 7 [.,[[[.,[.,.]],[[.,[.,.]],.]],.]] => 6 [.,[[[.,[[.,[.,.]],.]],[.,.]],.]] => 7 [.,[[[[.,[.,.]],[.,.]],[.,.]],.]] => 6 [.,[[[[.,[[[.,.],.],.]],.],.],.]] => 8 [.,[[[[[[.,.],.],[.,.]],.],.],.]] => 7 [.,[[[[[.,[.,[.,.]]],.],.],.],.]] => 8 [.,[[[[[.,[[.,.],.]],.],.],.],.]] => 8 [.,[[[[[[.,.],[.,.]],.],.],.],.]] => 7 [.,[[[[[[.,[.,.]],.],.],.],.],.]] => 8 [.,[[[[[[[.,.],.],.],.],.],.],.]] => 8 [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]] => 7 [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]] => 7 [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]] => 6 [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]] => 7 [[.,.],[.,[.,[.,[[[.,.],.],.]]]]] => 7 [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]] => 6 [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]] => 7 [[.,.],[[.,.],[[.,.],[[.,.],.]]]] => 5 [[.,.],[[.,.],[[.,[.,.]],[.,.]]]] => 5 [[.,.],[[.,.],[[.,[[.,.],.]],.]]] => 6 [[.,.],[[.,.],[[[[.,.],.],.],.]]] => 6 [[.,.],[[.,[.,.]],[.,[[.,.],.]]]] => 5 [[.,.],[[.,[.,.]],[[.,.],[.,.]]]] => 4 [[.,.],[[.,[.,.]],[[[.,.],.],.]]] => 5 [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]] => 5 [[.,.],[[.,[[.,.],.]],[[.,.],.]]] => 5 [[.,.],[[[[.,.],.],.],[[.,.],.]]] => 5 [[.,.],[[[[.,.],.],[.,.]],[.,.]]] => 5 [[.,.],[[[[.,.],[.,.]],.],[.,.]]] => 5 [[.,.],[[[[.,[.,.]],.],.],[.,.]]] => 6 [[.,.],[[.,[[.,.],[[.,.],.]]],.]] => 6 [[.,.],[[.,[[.,[[.,.],.]],.]],.]] => 7 [[.,.],[[[[.,.],.],[[.,.],.]],.]] => 5 [[.,.],[[[[.,.],[.,.]],[.,.]],.]] => 5 [[.,.],[[[[[[.,.],.],.],.],.],.]] => 7 [[.,[.,.]],[.,[[.,.],[[.,.],.]]]] => 5 [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]] => 5 [[.,[.,.]],[.,[[[[.,.],.],.],.]]] => 6 [[.,[.,.]],[[.,.],[.,[[.,.],.]]]] => 5 [[.,[.,.]],[[.,.],[[.,.],[.,.]]]] => 4 [[.,[.,.]],[[.,.],[[[.,.],.],.]]] => 5 [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]] => 4 [[.,[.,.]],[[[.,.],.],[[.,.],.]]] => 4 [[.,[.,.]],[[[.,.],[.,.]],[.,.]]] => 4 [[.,[.,.]],[[[.,[.,.]],.],[.,.]]] => 5 [[.,[.,.]],[[[[.,.],.],.],[.,.]]] => 5 [[.,[.,.]],[[.,[[.,.],[.,.]]],.]] => 5 [[.,[.,.]],[[.,[[.,[.,.]],.]],.]] => 6 [[.,[.,.]],[[[.,.],[[.,.],.]],.]] => 5 [[.,[.,.]],[[[.,[.,.]],[.,.]],.]] => 5 [[.,[.,.]],[[[[[.,.],.],.],.],.]] => 6 [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]] => 5 [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]] => 4 [[.,[.,[.,.]]],[.,[[[.,.],.],.]]] => 5 [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]] => 4 [[.,[.,[.,.]]],[[.,.],[[.,.],.]]] => 4 [[.,[.,[.,.]]],[[[.,.],.],[.,.]]] => 4 [[.,[[.,.],.]],[[.,.],[[.,.],.]]] => 4 [[.,[[.,.],.]],[[.,[.,.]],[.,.]]] => 4 [[.,[[.,.],.]],[[[.,.],.],[.,.]]] => 4 [[.,[[.,.],.]],[[.,[[.,.],.]],.]] => 5 [[.,[[.,.],.]],[[[.,.],[.,.]],.]] => 4 [[.,[[.,.],.]],[[[[.,.],.],.],.]] => 5 [[[.,.],[.,.]],[.,[[.,.],[.,.]]]] => 4 [[[.,.],[.,.]],[[.,.],[.,[.,.]]]] => 4 [[[[.,.],.],.],[[.,.],[[.,.],.]]] => 4 [[[[.,.],.],.],[[.,[.,.]],[.,.]]] => 4 [[[[.,.],.],.],[[[[.,.],.],.],.]] => 5 [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]] => 5 [[.,[.,[[.,.],.]]],[.,[[.,.],.]]] => 5 [[.,[[.,.],[.,.]]],[[.,.],[.,.]]] => 4 [[.,[[.,.],[.,.]]],[[.,[.,.]],.]] => 4 [[.,[[.,[.,.]],.]],[[.,.],[.,.]]] => 5 [[.,[[.,[.,.]],.]],[[.,[.,.]],.]] => 5 [[.,[[[.,.],.],.]],[[[.,.],.],.]] => 5 [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]] => 4 [[[.,.],[.,[.,.]]],[.,[[.,.],.]]] => 4 [[[.,.],[.,[.,.]]],[[.,[.,.]],.]] => 4 [[[.,.],[.,[.,.]]],[[[.,.],.],.]] => 4 [[[.,[.,.]],[.,.]],[[.,[.,.]],.]] => 4 [[[[.,.],.],[.,.]],[.,[[.,.],.]]] => 4 [[[[.,.],.],[.,.]],[[.,.],[.,.]]] => 4 [[[[.,.],.],[.,.]],[[[.,.],.],.]] => 4 [[[[.,.],[.,.]],.],[.,[[.,.],.]]] => 4 [[[[.,.],[.,.]],.],[[.,.],[.,.]]] => 4 [[[[.,.],[.,.]],.],[[[.,.],.],.]] => 4 [[[[.,[.,.]],.],.],[.,[[.,.],.]]] => 5 [[[[.,[.,.]],.],.],[[.,.],[.,.]]] => 5 [[[[.,[.,.]],.],.],[[[.,.],.],.]] => 5 [[.,[[.,.],[[.,.],.]]],[[.,.],.]] => 5 [[.,[[.,[[.,.],.]],.]],[[.,.],.]] => 6 [[[[.,.],.],[.,[.,.]]],[.,[.,.]]] => 4 [[[[.,.],.],[.,[.,.]]],[[.,.],.]] => 4 [[[[.,.],.],[[.,.],.]],[[.,.],.]] => 4 [[[[.,.],[.,.]],[.,.]],[.,[.,.]]] => 4 [[[[.,.],[.,.]],[.,.]],[[.,.],.]] => 4 [[[[.,[.,.]],.],[.,.]],[.,[.,.]]] => 5 [[[[.,[.,.]],.],[.,.]],[[.,.],.]] => 5 [[[[.,.],[.,[.,.]]],.],[.,[.,.]]] => 5 [[[[.,.],[.,[.,.]]],.],[[.,.],.]] => 5 [[[[.,.],[[.,.],.]],.],[[.,.],.]] => 5 [[[[.,[.,.]],[.,.]],.],[.,[.,.]]] => 5 [[[[.,[.,.]],[.,.]],.],[[.,.],.]] => 5 [[[[.,[.,[.,.]]],.],.],[.,[.,.]]] => 6 [[[[.,[[.,.],.]],.],.],[[.,.],.]] => 6 [[[[[[.,.],.],.],.],.],[[.,.],.]] => 6 [[.,[[.,[[.,[.,.]],.]],.]],[.,.]] => 7 [[.,[[[.,[.,.]],[.,.]],.]],[.,.]] => 6 [[[.,[.,.]],[[.,.],[.,.]]],[.,.]] => 4 [[[.,[.,.]],[[.,[.,.]],.]],[.,.]] => 5 [[[[.,.],.],[[.,.],[.,.]]],[.,.]] => 4 [[[[.,.],.],[[.,[.,.]],.]],[.,.]] => 5 [[[[.,.],.],[[[.,.],.],.]],[.,.]] => 5 [[[[.,.],[.,.]],[.,[.,.]]],[.,.]] => 4 [[[[.,.],[.,.]],[[.,.],.]],[.,.]] => 4 [[[[.,[.,.]],.],[[.,.],.]],[.,.]] => 5 [[[.,[[.,[.,.]],.]],[.,.]],[.,.]] => 6 [[[[.,.],[.,[.,.]]],[.,.]],[.,.]] => 5 [[[[.,.],[[.,.],.]],[.,.]],[.,.]] => 5 [[[[.,[.,.]],[.,.]],[.,.]],[.,.]] => 5 [[[[[[.,.],.],.],.],[.,.]],[.,.]] => 6 [[[.,[[.,[.,.]],[.,.]]],.],[.,.]] => 6 [[[[[[.,.],.],.],[.,.]],.],[.,.]] => 6 [[[[.,[[[.,.],.],.]],.],.],[.,.]] => 7 [[[[[[.,.],.],[.,.]],.],.],[.,.]] => 6 [[[[[.,[.,[.,.]]],.],.],.],[.,.]] => 7 [[[[[.,[[.,.],.]],.],.],.],[.,.]] => 7 [[[[[[.,.],[.,.]],.],.],.],[.,.]] => 6 [[[[[[.,[.,.]],.],.],.],.],[.,.]] => 7 [[[[[[[.,.],.],.],.],.],.],[.,.]] => 7 [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.] => 8 [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.] => 8 [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.] => 7 [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.] => 8 [[.,[.,[.,[.,[[[.,.],.],.]]]]],.] => 8 [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.] => 7 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[[[.,[.,.]],[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.] => 6 [[[.,.],[.,.]],[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]] => 7 [[[.,.],[.,.]],[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]] => 6 [[[.,[[.,.],[.,.]]],.],[.,[[.,[[.,.],[.,.]]],.]]] => 6 [[[.,[[.,.],[.,.]]],.],[.,[[[.,.],[.,.]],[.,.]]]] => 5 [[[[.,.],[.,.]],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]] => 6 [[[[.,.],[.,.]],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]] => 5 [[[.,[[.,[[.,.],[.,.]]],.]],.],[.,[[.,.],[.,.]]]] => 7 [[[.,[[[.,.],[.,.]],[.,.]]],.],[.,[[.,.],[.,.]]]] => 6 [[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[[.,.],[.,.]]] => 5 [[[[.,[[.,.],[.,.]]],.],[.,.]],[.,[[.,.],[.,.]]]] => 6 [[[[[.,.],[.,.]],[.,.]],[.,.]],[.,[[.,.],[.,.]]]] => 5 [[[.,[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]],.],[.,.]] => 10 [[[.,[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]],.],[.,.]] => 9 [[[.,[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.],[.,.]] => 7 [[[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]],.],[.,.]] => 9 [[[.,[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]],.],[.,.]] => 8 [[[[.,.],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]],[.,.]] => 7 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[.,[[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],.]] => 10 [.,[[.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]],.]] => 9 [.,[[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],.]] => 10 [.,[[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],.]] => 9 [.,[[[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]],.]] => 10 [.,[[[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]],.]] => 9 [.,[[[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]],.]] => 8 [.,[[[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]],.]] => 9 [.,[[[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]],.]] => 8 [.,[[[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]],.]] => 8 [.,[[[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]],.]] => 7 [.,[[[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]],.]] => 8 [.,[[[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]],.]] => 7 [.,[[[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]],.]] => 9 [.,[[[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]],.]] => 8 [.,[[[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]],.]] => 7 [.,[[[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]],.]] => 8 [.,[[[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]],.]] => 7 [.,[[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],.]] => 11 [.,[[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],.]] => 10 [.,[[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]],.]] => 9 [.,[[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],.]] => 10 [.,[[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],.]] => 9 [.,[[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]],.]] => 9 [.,[[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]],.]] => 8 [.,[[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]],.]] => 8 [.,[[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]],.]] => 7 [.,[[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],.]] => 10 [.,[[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],.]] => 9 [.,[[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]],.]] => 8 [.,[[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],.]] => 9 [.,[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],.]] => 8 [[.,[.,.]],[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]] => 10 [[.,[.,.]],[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]] => 9 [[.,[.,.]],[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]] => 8 [[.,[.,.]],[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]] => 9 [[.,[.,.]],[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]] => 8 [[.,[.,.]],[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]] => 8 [[.,[.,.]],[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]] => 7 [[.,[.,.]],[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]] => 7 [[.,[.,.]],[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]] => 6 [[.,[.,.]],[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]] => 9 [[.,[.,.]],[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]] => 8 [[.,[.,.]],[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]] => 7 [[.,[.,.]],[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]] => 8 [[.,[.,.]],[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]] => 7 [[.,[[.,[.,.]],.]],[[.,[[.,[[.,[.,.]],.]],.]],.]] => 8 [[.,[[.,[.,.]],.]],[[.,[[[.,[.,.]],[.,.]],.]],.]] => 7 [[.,[[.,[.,.]],.]],[[[.,[.,.]],[[.,[.,.]],.]],.]] => 6 [[.,[[.,[.,.]],.]],[[[.,[[.,[.,.]],.]],[.,.]],.]] => 7 [[.,[[.,[.,.]],.]],[[[[.,[.,.]],[.,.]],[.,.]],.]] => 6 [[[.,[.,.]],[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]] => 8 [[[.,[.,.]],[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]] => 7 [[[.,[.,.]],[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]] => 6 [[[.,[.,.]],[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]] => 7 [[[.,[.,.]],[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]] => 6 [[.,[[.,[[.,[.,.]],.]],.]],[[.,[[.,[.,.]],.]],.]] => 7 [[.,[[.,[[.,[.,.]],.]],.]],[[[.,[.,.]],[.,.]],.]] => 7 [[.,[[[.,[.,.]],[.,.]],.]],[[.,[[.,[.,.]],.]],.]] => 6 [[.,[[[.,[.,.]],[.,.]],.]],[[[.,[.,.]],[.,.]],.]] => 6 [[[.,[.,.]],[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]] => 6 [[[.,[.,.]],[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]] => 5 [[[.,[[.,[.,.]],.]],[.,.]],[[.,[[.,[.,.]],.]],.]] => 6 [[[.,[[.,[.,.]],.]],[.,.]],[[[.,[.,.]],[.,.]],.]] => 6 [[[[.,[.,.]],[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]] => 6 [[[[.,[.,.]],[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]] => 5 [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[[.,[.,.]],.]] => 9 [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[[.,[.,.]],.]] => 8 [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[[.,[.,.]],.]] => 7 [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[[.,[.,.]],.]] => 8 [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[[.,[.,.]],.]] => 7 [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]] => 7 [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]] => 6 [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[[.,[.,.]],.]] => 6 [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]] => 5 [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[[.,[.,.]],.]] => 8 [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[[.,[.,.]],.]] => 7 [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]] => 6 [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[[.,[.,.]],.]] => 7 [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]] => 6 [[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],[.,.]] => 11 [[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],[.,.]] => 10 [[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],[.,.]] => 9 [[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],[.,.]] => 10 [[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],[.,.]] => 9 [[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],[.,.]] => 9 [[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],[.,.]] => 8 [[.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]],[.,.]] => 8 [[.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]],[.,.]] => 7 [[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],[.,.]] => 10 [[.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]],[.,.]] => 9 [[.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]],[.,.]] => 8 [[.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]],[.,.]] => 9 [[.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]],[.,.]] => 8 [[[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]] => 9 [[[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]] => 8 [[[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]] => 7 [[[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]] => 8 [[[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]] => 7 [[[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]],[.,.]] => 7 [[[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]],[.,.]] => 6 [[[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]] => 7 [[[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]] => 6 [[[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]],[.,.]] => 8 [[[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]],[.,.]] => 7 [[[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]] => 6 [[[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]],[.,.]] => 7 [[[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]] => 6 [[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]],[.,.]] => 10 [[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]],[.,.]] => 9 [[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]],[.,.]] => 8 [[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]],[.,.]] => 9 [[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]],[.,.]] => 8 [[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]] => 8 [[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]] => 7 [[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]],[.,.]] => 7 [[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]] => 6 [[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]],[.,.]] => 9 [[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]],[.,.]] => 8 [[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]] => 7 [[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]],[.,.]] => 8 ----------------------------------------------------------------------------- Created: Mar 25, 2013 at 11:09 by Viviane Pons ----------------------------------------------------------------------------- Last Updated: Jan 17, 2018 at 23:05 by Martin Rubey