A005588 Number of free binary trees admitting height n.

2, 7, 52, 2133, 2590407, 3374951541062, 5695183504479116640376509, 16217557574922386301420514191523784895639577710480, 131504586847961235687181874578063117114329409897550318273792033024340388219235081096658023517076950

Offset: 1

Author

N. J. A. Sloane; entry revised by N. J. A. Sloane, Aug 31 2012

Keywords

• nonn
• easy
• core
• nice

Comments

a(n) is the number of free 3-trees which have a rooting as a binary tree of height n.

a(n) <= A002658(n+1) [Harary, et al.] "This is because any tree with a binary rooting of height h corresponds to a planted 3-tree of height h+1. [...] In general there are trees with more than one binary rooting of height h, so equality does not hold". - Michael Somos, Sep 02 2012

Examples

			+---------+
| o   o o | a(1) = 2
| |    \| |
| o     o |
+---------------------------------------------+
| o   o o     o   o o   o o   o o o   o o o o | a(2) = 7
| |    \|     |    \|   | |   |  \|    \| |/  |
| o     o   o o   o o   o o   o   o     o o   |
| |     |    \|    \|    \|    \ /       \|   |
| o     o     o     o     o     o         o   |
+---------------------------------------------+
a(3) = 52 while A002658(4) = 56 because there are 56 - 52 = 4 free binary trees admitting height 3 which have two rootings, while the rest have only one rooting. The four trees have degree sequences 32111, 322111, 3222111, 3321111. - _Michael Somos_, Sep 02 2012
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• David Wassermann, Table of n, a(n) for n = 1..12
• Harary, Frank; Palmer, Edgar M.; Robinson, Robert W., Counting free binary trees admitting a given height, J. Combin. Inform. System Sci. 17 (1992), no. 1-2, 175--181. MR1216977 (94c:05039)
• Harary, Frank; Palmer, Edgar M.; Robinson, Robert W., Counting free binary trees admitting a given height, J. Combin. Inform. System Sci. 17 (1992), no. 1-2, 175-181. (Annotated scanned copy)
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees
• Index entries for "core" sequences

Crossrefs

Cf. A002658, A006894.

Programs

Mathematica

bin2[n_] = Binomial[n, 2];
bin3[n_] = Binomial[n, 3];
p[0] = q[0] = 0;
p[1] = q[1] = 1;
q[h1_] := q[h1] = With[{h = h1-1}, q[h] + p[h]];
p[h1_] := p[h1] = With[{h = h1-1}, bin2[1 + p[h]] + p[h] q[h]];
a[h_] := a[h] = bin3[2 + p[h]] + bin2[1 + p[h]] q[h];
b[h_] := b[h] = bin2[1 + p[h]];
e[h_, i_] := e[h, i] = 1 + Sum[d[j, i], {j, h-1}];
d[h_, h_] := 0; d[h_, i_] := p[h] /; i > h;
d[h1_, i1_] := d[h1, i1] = With[{h = h1-1, i = i1-1}, bin2[1 + d[h, i]] + d[h, i] e[h, i]]; d[h_, 1] := d[h, 1] = p[h] - p[h-1];
e[h_, 1] := e[h, 1] = p[h-1];
t1[h_] := Sum[a[h-i] - bin3[2 + d[h-i, i]] - bin2[1 + d[h-i, i]] e[h-i, i], {i, Quotient[h, 2]}];
t2[h_] := Sum[b[h-i+1] - bin2[1 + d[h-i+1, i]], {i, Quotient[h+1, 2]}];
t[h_] := bin2[1 + p[h]] + t1[h] + t2[h];
Table[t[n], {n, 1, 12}] (* Jean-François Alcover, Apr 22 2013, program corrected and improved by Michael Somos *)

Formula

Harary et al. give a complicated recurrence.