A217710 Cardinality of the set of possible heights of AVL trees with n (leaf-) nodes.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Author

Alois P. Heinz, Mar 20 2013

Keywords

• nonn

Comments

a(n) increases at Fibonacci numbers (A000045) and decreases at powers of 2 plus 1 (A000051) for n>=8.

a(n) is the height (number of nonzero elements) of column n of triangles A143897, A217298.

Examples

			a(8) = 2: We have 1 AVL tree with n=8 (leaf-) nodes of height 3 and 16 of height 4 (8 is both Fibonacci number and power of 2):
           o              o
         /   \          /   \
       o       o      o       o
      / \     / )    / \     / \
     o   o   o  N   o   o   o   o
    / ) ( ) ( )    ( ) ( ) ( ) ( )
   o  N N N N N    N N N N N N N N
  ( )
  N N
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50000
• Wikipedia, AVL tree
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A000045, A000051, A000079, A029837, A072649, A143897, A217298.

Programs

Maple

a:= proc(n) local j, p; for j from ilog[(1+sqrt(5))/2](n)
       while combinat[fibonacci](j+1)<=n do od;
       p:= ilog2(n);
       j-p-`if`(2^p issqr(t+4) or issqr(t-4))(5*n^2), 1, 0)-
     `if`((t-> is(2^ilog2(t)=t))(n-1), 1, 0))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Aug 14 2021

Mathematica

a[n_] := Module[{j, p}, For[j = Log[(1+Sqrt[5])/2, n] // Floor, Fibonacci[j+1] <= n, j++]; p = Log[2, n] // Floor; j-p-If[2^p < n, 2, 1]]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

Formula

a(n) = A072649(n) - A029837(n).