A143897 Triangle read by rows: T(n,k) = number of AVL trees of height n with k (leaf-) nodes, n>=0, fibonacci(n+2)<=k<=2^n.

1, 1, 2, 1, 4, 6, 4, 1, 16, 32, 44, 60, 70, 56, 28, 8, 1, 128, 448, 864, 1552, 2720, 4288, 6312, 9004, 11992, 14372, 15400, 14630, 11968, 8104, 4376, 1820, 560, 120, 16, 1, 4096, 22528, 67584, 159744, 334080, 644992, 1195008, 2158912, 3811904, 6617184

Offset

0,3

References

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 239, Eq 79, A_5.

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Links

Alois P. Heinz, Rows n = 0..10, flattened

Ralf Hinze, Functional Pearls: Purely functional 1-2 brother trees, Journal of Functional Programming, 19(6):633-644, 2009, DOI: 10.1017/S0956796809007333.

R. C. Richards, Shape distribution of height-balanced trees, Info. Proc. Lett., 17 (1983), 17-20.

Wikipedia, AVL tree

Index entries for sequences related to rooted trees

Formula

See program.

Example

There are 2 AVL trees of height 2 with 3 (leaf-) nodes:
       o       o
      / \     / \
     o   N   N   o
    / \         / \
   N   N       N   N
Triangle begins:
  1
  . 1
  . . 2 1
  . . . . 4 6 4  1
  . . . . . . . 16 32 44 60 70  56  28   8    1
  . . . . . . .  .  .  .  .  . 128 448 864 1552 2720 ...

Maple

T:= proc(n, k) local B, z; B:= proc(x, y, d) if d>=1 then x+B(x^2+2*x*y, x, d-1) else x fi end; if n=0 then if k=1 then 1 else 0 fi else coeff(B(z, 0, n), z, k) -coeff(B(z, 0, n-1), z, k) fi end: fib:= m-> (Matrix([[1, 1], [1, 0]])^m)[1, 2]: seq(seq(T(n, k), k=fib(n+2)..2^n), n=0..6);

Mathematica

t[n_, k_] := Module[{ b, z }, b[x_, y_, d_] := If[d >= 1, x + b[x^2 + 2*x*y, x, d-1], x]; If[n == 0, If[k == 1, 1, 0], Coefficient[b[z, 0, n], z, k] - Coefficient [b[z, 0, n-1], z, k]]]; fib[m_] := MatrixPower[{{1, 1}, {1, 0}}, m][[1, 2]]; Table[Table[t[n, k], {k, fib[n+2], 2^n}], {n, 0, 6}] // Flatten (* Jean-François Alcover, Dec 05 2013, translated from Alois P. Heinz's Maple program *)

Crossrefs

Row sums give: A029758.

Column sums give: A006265.

Column sums of first 5 or 6 rows give: A036662 or A134306.

First elements of rows give: A174677.

First, last elements of columns give: A217299, A217300.

Row lengths give: 1+A008466(n).

Column heights give: A217710(k).

Triangle read by columns gives: A217298.

Cf. A000045, A000079.

Sequence in context: A343964 A345135 A291057 * A217298 A217299 A333340

Adjacent sequences: A143894 A143895 A143896 * A143898 A143899 A143900

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Sep 04 2008

Status

approved