A316944 Total height of the binary search trees summed over all permutations of [n].

0, 1, 4, 16, 80, 456, 3072, 23536, 202304, 1937920, 20470400, 236172288, 2955465216, 39893618688, 577937479680, 8944476580864, 147277509541888, 2570740210700288, 47418288632692736, 921669646969167872, 18829500772271472640, 403390045252173381632

Offset: 0

Author

Alois P. Heinz, Jul 17 2018

Keywords

• nonn

Comments

Each permutation of [n] generates a binary search tree of height h (floor(log_2(n))+1 <= h <= n) when its elements are inserted in that order into the initially empty tree. The average height of a binary search tree on n elements is A195582(n)/A195583(n).

Empty external nodes are counted in determining the height of a search tree.

Examples

			a(3) = 16 = 3 + 3 + (2+2) + 3 + 3:
.
          3         3        2        1         1
         / \       / \      / \      / \       / \
        2   o     1   o    1   3    o   3     o   2
       / \       / \      ( ) ( )      / \       / \
      1   o     o   2     o o o o     2   o     o   3
     / \           / \               / \           / \
    o   o         o   o   (2,1,3)   o   o         o   o
     (3,2,1)     (3,1,2)  (2,3,1)    (1,3,2)   (1,2,3)
.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..449
• Wikipedia, Binary search tree
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A000142, A000523, A195581, A195582, A195583, A244108, A335921.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, `if`(k add(k*(b(n, k)-b(n, k-1)), k=0..n):
seq(a(n), n=0..25);

Mathematica

b[n_, k_] := b[n, k] = If[n < 2, If[k < n, 0, 1], Sum[Binomial[n - 1, r]* b[r, k - 1] b[n - 1 - r, k - 1], {r, 0, n - 1}]];
a[n_] := Sum[k(b[n, k] - b[n, k - 1]), {k, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A195581(n,k) = Sum_{k=0..n} k * A244108(n,k).

a(n) = A000142(n) * A195582(n)/A195583(n).