A005588 - OEIS

%I M1813 #68 Aug 28 2022 08:22:55

%S 2,7,52,2133,2590407,3374951541062,5695183504479116640376509,

%T 16217557574922386301420514191523784895639577710480,

%U 131504586847961235687181874578063117114329409897550318273792033024340388219235081096658023517076950

%N Number of free binary trees admitting height n.

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

%C 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

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

%H David Wassermann, <a href="/A005588/b005588.txt">Table of n, a(n) for n = 1..12</a>

%H Harary, Frank; Palmer, Edgar M.; Robinson, Robert W., <a href="http://cobweb.cs.uga.edu/~rwr/publications/binary.pdf">Counting free binary trees admitting a given height</a>, J. Combin. Inform. System Sci. 17 (1992), no. 1-2, 175--181. MR1216977 (94c:05039)

%H Harary, Frank; Palmer, Edgar M.; Robinson, Robert W., <a href="/A005588/a005588.pdf">Counting free binary trees admitting a given height</a>, J. Combin. Inform. System Sci. 17 (1992), no. 1-2, 175-181. (Annotated scanned copy)

%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>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F Harary et al. give a complicated recurrence.

%e +---------+

%e | o   o o | a(1) = 2

%e | |    \| |

%e | o     o |

%e +---------------------------------------------+

%e | o   o o     o   o o   o o   o o o   o o o o | a(2) = 7

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

%e | o     o   o o   o o   o o   o   o     o o   |

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

%e | o     o     o     o     o     o         o   |

%e +---------------------------------------------+

%e 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

%t bin2[n_] = Binomial[n, 2];

%t bin3[n_] = Binomial[n, 3];

%t p[0] = q[0] = 0;

%t p[1] = q[1] = 1;

%t q[h1_] := q[h1] = With[{h = h1-1}, q[h] + p[h]];

%t p[h1_] := p[h1] = With[{h = h1-1}, bin2[1 + p[h]] + p[h] q[h]];

%t a[h_] := a[h] = bin3[2 + p[h]] + bin2[1 + p[h]] q[h];

%t b[h_] := b[h] = bin2[1 + p[h]];

%t e[h_, i_] := e[h, i] = 1 + Sum[d[j, i], {j, h-1}];

%t d[h_, h_] := 0; d[h_, i_] := p[h] /; i > h;

%t 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];

%t e[h_, 1] := e[h, 1] = p[h-1];

%t 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]}];

%t t2[h_] := Sum[b[h-i+1] - bin2[1 + d[h-i+1, i]], {i, Quotient[h+1, 2]}];

%t t[h_] := bin2[1 + p[h]] + t1[h] + t2[h];

%t Table[t[n], {n, 1, 12}] (* _Jean-François Alcover_, Apr 22 2013, program corrected and improved by _Michael Somos_ *)

%Y Cf. A002658, A006894.

%K nonn,easy,core,nice

%O 1,1

%A _N. J. A. Sloane_; entry revised by _N. J. A. Sloane_, Aug 31 2012