A244108 -id:A244108 - cp's OEIS frontend

A195581 Number T(n,k) of permutations of {1,2,...,n} that result in a binary search tree of height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 16, 8, 0, 0, 0, 40, 64, 16, 0, 0, 0, 80, 400, 208, 32, 0, 0, 0, 80, 2240, 2048, 608, 64, 0, 0, 0, 0, 11360, 18816, 8352, 1664, 128, 0, 0, 0, 0, 55040, 168768, 104448, 30016, 4352, 256, 0, 0, 0, 0, 253440, 1508032, 1277568, 479040, 99200, 11008, 512

Offset: 0

Alois P. Heinz, Sep 20 2011

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

			T(3,3) = 4, because 4 permutations of {1,2,3} result in a binary search tree of height 3:
  (1,2,3):   1       (1,3,2):   1     (3,1,2):   3   (3,2,1):   3
            / \                / \              / \            / \
           o   2              o   3            1   o          2   o
              / \                / \          / \            / \
             o   3              2   o        o   2          1   o
                / \            / \              / \        / \
               o   o          o   o            o   o      o   o
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 2,  4;
  0, 0, 0, 16,      8;
  0, 0, 0, 40,     64,      16;
  0, 0, 0, 80,    400,     208,      32;
  0, 0, 0, 80,   2240,    2048,     608,     64;
  0, 0, 0,  0,  11360,   18816,    8352,   1664,   128;
  0, 0, 0,  0,  55040,  168768,  104448,  30016,  4352,   256;
  0, 0, 0,  0, 253440, 1508032, 1277568, 479040, 99200, 11008, 512;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A000142. Column sums give A227822.
Main diagonal gives A011782, lower diagonal gives A076616.
T(n,A000523(n)+1) = A076615(n).
T(2^n-1,n) = A056972(n).
T(2n,n) = A265846(n).
Cf. A195582, A195583, A244108 (the same read by columns), A316944, A317012.

b:= proc(n, k) option remember; `if`(n<2, `if`(k b(n, k)-b(n, k-1):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, k_] := b[n, k] = If[n == 0, 1, If[n == 1, If[k > 0, 1, 0], Sum[Binomial[n-1, r-1]*b[r-1, k-1]*b[n-r, k-1], {r, 1, n}] ] ]; t [n_, k_] := b[n, k] - If[k > 0, b[n, k-1], 0]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

Sum_{k=0..n} k * T(n,k) = A316944(n).

Sum_{k=n..2^n-1} k * T(k,n) = A317012(n).

A195582 Numerator of the average height of a binary search tree on n elements.

Original entry on oeis.org

0, 1, 2, 8, 10, 19, 64, 1471, 3161, 3028, 6397, 27956, 58307, 168652, 190031, 794076401, 817191437, 57056556523, 65776878541, 112508501827291, 32836043478431, 24620974441660973, 30663050241335933, 280904716386831931, 1713934856212591039, 12438570098319186469

Offset: 0

Alois P. Heinz, Sep 20 2011

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

			0/1, 1/1, 2/1, 8/3, 10/3, 19/5, 64/15, 1471/315, 3161/630, 3028/567, 6397/1134, 27956/4725, 58307/9450, 168652/26325, 190031/28665 ... = A195582/A195583
For n = 3 there are 2 permutations of {1,2,3} resulting in a binary search tree of height 2 and 4 permutations resulting in a tree of height 3.  The average height is (2*2+4*3)/3! = (4+12)/6 = 16/6 = 8/3.

Alois P. Heinz, Table of n, a(n) for n = 0..478
B. Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Denominators: A195583.

Cf. A000142, A195581, A195596, A195597, A195598, A195599, A195600, A195601, A244108, A316944.

b:= proc(n,k) option remember;
      if n=0 then 1
    elif n=1 then `if`(k>0, 1, 0)
    else add(binomial(n-1,r-1) *b(r-1,k-1) *b(n-r,k-1), r=1..n)
      fi
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
a:= n-> add(T(n,k)*k, k=0..n)/n!:
seq(numer(a(n)), n=0..30);

b[n_, k_] := b[n, k] = If[n==0, 1, If[n==1, If[k>0, 1, 0], Sum[Binomial[n - 1, r-1]*b[r-1, k-1]*b[n-r, k-1], {r, 1, n}]]]; T[n_, k_] := b[n, k] - If[ k>0, b[n, k-1], 0]; a[n_] := Sum[T[n, k]*k, {k, 0, n}]/n!; Table[ Numerator[a[n]], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

A195582(n)/A195583(n) = 1/n! * Sum_{k=1..n} k * A195581(n,k).
A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with alpha = 4.311... (A195596) and beta = 1.953... (A195599).
A195582(n)/A195583(n) = A316944(n) / A000142(n).

A076615 Number of permutations of {1,2,...,n} that result in a binary search tree with the minimum possible height.

Original entry on oeis.org

1, 1, 2, 2, 16, 40, 80, 80, 11360, 55040, 253440, 1056000, 3801600, 10982400, 21964800, 21964800, 857213660160, 7907423180800, 72155129446400, 645950912921600, 5622693241651200, 47110389109555200, 375570435981312000, 2811021970538496000, 19445103757787136000

Offset: 0

Jeffrey Shallit, Oct 22 2002

nonn

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

			a(3) = 2 because only the permutations (2,1,3) and (2,3,1) result in a binary search tree of minimal height. In both cases you will get the following binary search tree:
        2
      /   \
     1     3
    / \   / \
   o   o o   o

Alois P. Heinz, Table of n, a(n) for n = 0..511 (first 50 terms from Michal Forisek)
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Leftmost nonzero elements in rows of A195581, A244108.

Cf. A076616, A328349.

b:= proc(n,k) option remember;
      if n=0 then 1
    elif n=1 then `if`(k>0, 1, 0)
    else add(binomial(n-1, r-1) *b(r-1, k-1) *b(n-r, k-1), r=1..n)
      fi
    end:
a:= n-> b(n, ilog2(n)+1):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 20 2011

b[n_, k_] := b[n, k] = Which[n==0, 1, n==1, If[k>0, 1, 0], True, Sum[ Binomial[n-1, r-1]*b[r-1, k-1]*b[n-r, k-1], {r, 1, n}]]; a[n_] := b[n, Floor @ Log[2, n]+1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 19 2017, after Alois P. Heinz *)

from math import factorial as f, log, floor
B= {}
def b(n,x):
    if (n,x) in B: return B[(n,x)]
    if n<=1: B[(n,x)] = int(x>=0)
    else: B[(n,x)]=sum([b(i,x-1)*b(n-1-i,x-1)*f(n-1)//f(i)//f(n-1-i) for i in range(n)])
    return B[(n,x)]
for n in range(1,51): print(b(n,floor(log(n,2))))
# Michal Forisek, Sep 19 2011

Let b(n,k) be the number of permutations of {1,2,...,n} that produce a binary search tree of depth at most k. We have:
b(0,k) = 1 for k>=0,
b(1,0) = 0,
b(1,k) = 1 for k>0,
b(n,k) = Sum_{r=1..n} C(n-1,r-1) * b(r-1,k-1) * b(n-r,k-1) for n>=2, k>=0.
Then a(n) = b(n,floor(log_2(n))+1).

Extended beyond a(8) and efficient formula by Michal Forisek, Sep 19 2011

a(0)=1 prepended by Alois P. Heinz, Jul 18 2018

A076616 Number of permutations of {1,2,...,n} that result in a binary search tree (when elements of the permutation are inserted in that order) of height n-1 (i.e., the second largest possible height).

Original entry on oeis.org

0, 0, 0, 2, 16, 64, 208, 608, 1664, 4352, 11008, 27136, 65536, 155648, 364544, 843776, 1933312, 4390912, 9895936, 22151168, 49283072, 109051904, 240123904, 526385152, 1149239296, 2499805184, 5419040768, 11710496768, 25232932864, 54223962112, 116232552448

Offset: 0

Jeffrey Shallit, Oct 22 2002

			a(3) = 2 because only the permutations (2,1,3) and (2,3,1) result in a search tree of height 2 (notice we count empty external nodes in determining the height). The largest such trees are of height 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Lower diagonal of A195581 or of A244108.

Cf. A049611, A076615, A084851, A100312.

a:= n-> max(-(<<0|1|0>, <0|0|1>, <8|-12|6>>^n. <<1/2, 1, 1>>)[1$2], 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 20 2011

Join[{0, 0, 0}, LinearRecurrence[{6, -12, 8}, {2, 16, 64}, 40]] (* Jean-François Alcover, Jan 09 2025 *)

concat(vector(3), Vec(2*x^3*(1+2*x-4*x^2)/(1-2*x)^3 + O(x^50))) \\ Colin Barker, May 16 2016

G.f.: 2*(4*x^2-2*x-1)*x^3/(2*x-1)^3. - Alois P. Heinz, Sep 20 2011
From Colin Barker, May 16 2016: (Start)
a(n) = 2^(n-3)*(n^2-n-4) for n>2.
a(n) = 6*a(n-1)-12*a(n-2)+8*a(n-3) for n>5.
(End)
From Alois P. Heinz, May 31 2022: (Start)
a(n) = 2 * A100312(n-3) for n>=3.
a(n) = 16 * A049611(n-3) = 16 * A084851(n-4) for n>=4. (End)

More terms from Alois P. Heinz, Sep 20 2011

A335919 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), n>=0, max(0,floor(log_2(n))+1)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 8, 6, 20, 16, 4, 40, 56, 32, 1, 68, 152, 144, 64, 94, 376, 480, 352, 128, 114, 844, 1440, 1376, 832, 256, 116, 1744, 4056, 4736, 3712, 1920, 512, 94, 3340, 10856, 15248, 14272, 9600, 4352, 1024, 60, 5976, 27672, 47104, 50784, 40576, 24064

Offset: 0

Alois P. Heinz, Jun 29 2020

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

T(n,k) is defined for n,k >= 0. The triangle contains only the positive terms. Terms not shown are zero.

			Triangle T(n,k) begins:
  1;
     1;
        2;
        1, 4;
           6,   8;
           6,  20,   16;
           4,  40,   56,   32;
           1,  68,  152,  144,   64;
               94,  376,  480,  352,  128;
              114,  844, 1440, 1376,  832,  256;
              116, 1744, 4056, 4736, 3712, 1920, 512;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A000108.
Column sums give A001699.
Main diagonal gives A011782.
T(n+3,n+2) gives A014480.
T(n,max(0,A000523(n)+1)) = A328349(n).
Cf. A073345, A073429 (another version with 0's), A076615, A195581, A244108, A335920 (the same read by columns), A335921, A335922.

g:= n-> `if`(n=0, 0, ilog2(n)+1):
b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
seq(seq(T(n, k), k=g(n)..n), n=0..12);

g[n_] := If[n == 0, 0, Floor@Log[2, n]+1];
b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
Table[Table[T[n, k], {k, g[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A335921(n).

Sum_{n=k..2^k-1} n * T(n,k) = A335922(k).

A335920 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 6, 4, 1, 8, 20, 40, 68, 94, 114, 116, 94, 60, 28, 8, 1, 16, 56, 152, 376, 844, 1744, 3340, 5976, 10040, 15856, 23460, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1, 32, 144, 480, 1440, 4056

Offset: 0

Alois P. Heinz, Jun 29 2020

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

T(n,k) is defined for n,k >= 0. The triangle contains only the positive terms. Terms not shown are zero.

			Triangle T(n,k) begins:
  1;
     1;
        2;
        1, 4;
           6,   8;
           6,  20,   16;
           4,  40,   56,   32;
           1,  68,  152,  144,   64;
               94,  376,  480,  352,  128;
              114,  844, 1440, 1376,  832,  256;
              116, 1744, 4056, 4736, 3712, 1920, 512;
  ...

Alois P. Heinz, Columns k = 0..10, flattened
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Row sums give A000108.
Column sums give A001699.
Main diagonal gives A011782.
T(n+3,n+2) gives A014480.
T(n,max(0,A000523(n)+1)) = A328349(n).
Cf. A073345, A076615, A195581, A244108, A335919 (the same read by rows), A335921, A335922.

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

b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
Table[Table[T[n, k], {n, k, 2^k - 1}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

Sum_{k=0..n} k * T(n,k) = A335921(n).

Sum_{n=k..2^k-1} n * T(n,k) = A335922(k).

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

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Jul 17 2018

nonn

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.

			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)
.

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

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

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);

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 *)

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).

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

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Jul 18 2018

nonn

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

			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)
.

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

Cf. A195581, A227822, A244108, A335922.

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);

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 *)

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

A227822 Number of permutations of [n], [n+1], ... that result in a binary search tree of height n.

Original entry on oeis.org

1, 1, 4, 220, 60092152, 203720181459953921762400, 7088043372247785801830314829178419617696182324188730917543544992

Offset: 0

Alois P. Heinz, Jul 31 2013

nonn

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

			a(2) = 4, because 4 permutations of {1,2}, {1,2,3}, ... result in a binary search tree of height 2:
  (1,2):   1      (2,1):   2    (2,1,3), (2,3,1):    2
          / \             / \                      /   \
         o   2           1   o                    1     3
            / \         / \                      / \   / \
           o   o       o   o                    o   o o   o

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

Column sums of A195581 and of A244108.

Cf. A317012.

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

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[b[k, n] - b[k, n - 1], {k, n, 2^n - 1}];
a /@ Range[0, 6] (* Jean-François Alcover, Apr 02 2021, after Alois P. Heinz *)

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

Terms corrected by Alois P. Heinz, Dec 08 2015

A265846 Number of permutations of {1,2,...,2n} that result in a binary search tree of height n.

Original entry on oeis.org

1, 0, 0, 80, 11360, 1508032, 197163648, 27624399104, 4256968553472, 724948555548672, 136034167652859904, 27976811931437752320, 6269253131872946298880, 1522110257603797873737728, 398311795891656532955725824, 111813044381693547723191418880

Offset: 0

Alois P. Heinz, Dec 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Binary search tree
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A195581, A244108.

b:= proc(n, k) option remember; `if`(n<2, `if`(k b(2*n, n)-b(2*n, n-1):
seq(a(n), n=0..20);

b[n_, k_] := b[n, k] = If[n<2, If[kJean-François Alcover, Feb 25 2017, translated from Maple *)

a(n) = A195581(2n,n) = A244108(2n,n).

a(n) ~ c * d^n * (n-1)!, where d = -16*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 19.643259858273023595006139220961408..., c = 1/(2^(5/2)*Pi*sqrt(-1 - LambertW(-1, -exp(-1/2)/2))) = 0.0646979349409396546649864277836... . - Vaclav Kotesovec, Jan 02 2016, updated Mar 17 2024 and May 14 2025

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A195581 Number T(n,k) of permutations of {1,2,...,n} that result in a binary search tree of height k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A195582 Numerator of the average height of a binary search tree on n elements.

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A076615 Number of permutations of {1,2,...,n} that result in a binary search tree with the minimum possible height.

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A076616 Number of permutations of {1,2,...,n} that result in a binary search tree (when elements of the permutation are inserted in that order) of height n-1 (i.e., the second largest possible height).

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A335919 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), n>=0, max(0,floor(log_2(n))+1)<=k<=n, read by rows.

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A335920 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

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

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