A330627 Number of non-isomorphic phylogenetic trees with n nodes.

0, 1, 1, 1, 2, 2, 4, 5, 9, 14, 24, 39, 69, 116, 205, 357, 632, 1118, 2001, 3576, 6445, 11627, 21080, 38293, 69819, 127539, 233644, 428825, 788832, 1453589, 2683602, 4962167, 9190155, 17044522, 31655676, 58866237, 109600849, 204293047, 381212823, 712073862

Offset

1,5

Comments

A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets. Each branching as well as each element of each leaf contributes to the number of nodes.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

G.f.: A(x) satisfies A(x) = x*(1/(1-x) - A(x) - 2 + exp(Sum_{k>0} A(x^k)/k)). - Andrew Howroyd, Jan 02 2021

Example

Non-isomorphic representatives of the a(2) = 1 through a(9) = 9 trees (commas and outer brackets elided):
  1  12  123  1234    12345    123456     1234567      12345678
              (1)(2)  (1)(23)  (1)(234)   (1)(2345)    (1)(23456)
                               (12)(34)   (12)(345)    (12)(3456)
                               (1)(2)(3)  (1)(2)(34)   (123)(456)
                                          (1)((2)(3))  (1)(2)(345)
                                                       (1)(23)(45)
                                                       (1)((2)(34))
                                                       (1)(2)(3)(4)
                                                       (12)((3)(4))

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, EulerT(v)[n] - v[n] + 1)); v} \\ Andrew Howroyd, Jan 02 2021

Crossrefs

Phylogenetic trees by number of labels are A005804, with unlabeled version A141268.

Balanced phylogenetic trees are A320154.

Cf. A000311, A000669, A001678, A004114, A005121, A007716, A048816, A060356, A330465, A330467, A330469.

Sequence in context: A007147 A230380 A127968 * A188541 A037026 A116651

Adjacent sequences: A330624 A330625 A330626 * A330628 A330629 A330630

Keyword

nonn

Author

Gus Wiseman, Dec 28 2019

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2021

Status

approved