A317012 Total number of elements in all permutations of [n], [n+1], ... that result in a binary search tree of height n.

0, 1, 10, 1316, 840124672, 6110726696100443604557936, 439451426203104201222708341688051362423089952907263634936946272224

Offset: 0

Author

Alois P. Heinz, Jul 18 2018

Keywords

• nonn

Comments

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

Examples

			a(2) = 10 = 2 + 3 + 3 + 2:
.
         2           2           1
        / \         / \         / \
       1   o       1   3       o   2
      / \         ( ) ( )         / \
     o   o        o o o o        o   o
      (2,1)   (2,1,3) (2,3,1)   (1,2)
.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..9
• Wikipedia, Binary search tree
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A195581, A227822, A244108, A335922.

Programs

Maple

b:= proc(n, k) option remember; `if`(n<2, `if`(k b(n, k)-b(n, k-1):
a:= k-> add(T(n, k)*n, n=k..2^k-1):
seq(a(n), n=0..6);

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}]];
T[n_, k_] := b[n, k] - b[n, k-1];
a[k_] := Sum[T[n, k] n, {n, k, 2^k - 1}];
a /@ Range[0, 6] (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=n..2^n-1} k * A195581(k,n) = Sum_{k=n..2^n-1} k * A244108(k,n).