A316944 - OEIS

A316944

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

%I #18 Jan 27 2021 10:13:41

%S 0,1,4,16,80,456,3072,23536,202304,1937920,20470400,236172288,

%T 2955465216,39893618688,577937479680,8944476580864,147277509541888,

%U 2570740210700288,47418288632692736,921669646969167872,18829500772271472640,403390045252173381632

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

%C 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).

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

%H Alois P. Heinz, <a href="/A316944/b316944.txt">Table of n, a(n) for n = 0..449</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_search_tree">Binary search tree</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

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

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

%e a(3) = 16 = 3 + 3 + (2+2) + 3 + 3:

%e .

%e 3 3 2 1 1

%e / \ / \ / \ / \ / \

%e 2 o 1 o 1 3 o 3 o 2

%e / \ / \ ( ) ( ) / \ / \

%e 1 o o 2 o o o o 2 o o 3

%e / \ / \ / \ / \

%e o o o o (2,1,3) o o o o

%e (3,2,1) (3,1,2) (2,3,1) (1,3,2) (1,2,3)

%e .

%p b:= proc(n, k) option remember; `if`(n<2, `if`(k<n, 0, 1),

%p add(binomial(n-1, r)*b(r, k-1)*b(n-1-r, k-1), r=0..n-1))

%p end:

%p a:= n-> add(k*(b(n, k)-b(n, k-1)), k=0..n):

%p seq(a(n), n=0..25);

%t 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 a[n_] := Sum[k(b[n, k] - b[n, k - 1]), {k, 0, n}];

%t a /@ Range[0, 25] (* _Jean-François Alcover_, Jan 27 2021, after _Alois P. Heinz_ *)

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

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 17 2018