A318185 -id:A318185 - cp's OEIS frontend

A290689 Number of transitive rooted trees with n nodes.

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 143, 229, 370, 592, 955, 1527, 2457, 3929, 6304, 10081

Offset: 1

Gus Wiseman, Oct 19 2017

A rooted tree is transitive if every proper terminal subtree is also a branch of the root. First differs from A206139 at a(13) = 143.

Regarding the notation, a rooted tree is a finite multiset of rooted trees. For example, the rooted tree (o(o)(oo)) is short for {{},{{}},{{},{}}}. Each "o" is a leaf. Each pair of parentheses corresponds to a non-leaf node (such as the root). Its contents "(...)" represent a branch. - Gus Wiseman, Nov 16 2024

			The a(7) = 8 7-node transitive rooted trees are: (o(oooo)), (oo(ooo)), (o(o)((o))), (o(o)(oo)), (ooo(oo)), (oo(o)(o)), (oooo(o)), (oooooo).

Robert P. P. McKone, The transitive rooted trees with n=1 to n=22 nodes.

The restriction to identity trees (A004111) is A279861, ranks A290760.
These trees are ranked by A290822.
The anti-transitive version is A306844, ranks A324758.
The totally transitive case is A318185 (by leaves A318187), ranks A318186.
A version for integer partitions is A324753, for subsets A324736.
The ordered version is A358453, ranks A358457, undirected A358454.
Cf. A000081, A001192, A001678, A324770, A324969, A325705, A358455, A358460.

nn=18;
rtall[n_]:=If[n===1,{{}},Module[{cas},Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])]]];
Table[Length[Select[rtall[n],Complement[Union@@#,#]==={}&]],{n,nn}]

a(20) from Robert Price, Sep 13 2018

a(21)-a(22) from Robert P. P. McKone, Dec 16 2023

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882

Offset: 1

Gus Wiseman, Mar 17 2019

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (o = {}):
  {{{{{{{o}}}}}}}
  {{{{{o,{o}}}}}}
  {{{{o,{{o}}}}}}
  {{{o,{{{o}}}}}}
  {{{o,{o,{o}}}}}
  {{{{o},{{o}}}}}
  {{o,{{{{o}}}}}}
  {{o,{{o,{o}}}}}
  {{o,{o,{{o}}}}}
  {{{o},{{{o}}}}}
  {{{o},{o,{o}}}}
  {{o,{o},{{o}}}}
  {o,{{{{{o}}}}}}
  {o,{{{o,{o}}}}}
  {o,{{o,{{o}}}}}
  {o,{{o},{{o}}}}
  {{o},{{{{o}}}}}
  {{o},{{o,{o}}}}
  {{o},{o,{{o}}}}
  {{{o}},{o,{o}}}

			The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
  o  (o)  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o(o(o))))
                          (o((o)))   ((o((o))))   (o((o(o))))
                          ((((o))))  (o(((o))))   ((((o(o)))))
                                     (((((o)))))  (((o)((o))))
                                                  (((o((o)))))
                                                  ((o)(((o))))
                                                  ((o(((o)))))
                                                  (o((((o)))))
                                                  ((((((o))))))

Gus Wiseman, The a(9) = 41 anti-transitive rooted identity trees.

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.

Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A324765 Number of recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 26, 52, 119, 266, 618, 1432, 3402, 8093, 19505, 47228, 115244, 282529, 696388, 1723400

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.

			The a(1) = 1 through a(6) = 11 recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   (o((oo)))
                          ((((o))))  (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Cf. A000081, A290689, A301700, A304360, A306844, A317787, A318185.
Cf. A324695, A324751, A324756, A324758, A324764, A324766, A324767, A324768.
Cf. A324838, A324840, A324844.

nallt[n_]:=Select[Union[Sort/@Join@@(Tuples[nallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@#,#]=={}&];
Table[Length[nallt[n]],{n,10}]

A324844 Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 32, 71, 170, 406, 1002, 2469, 6204, 15644, 39871, 102116, 263325, 682079, 1775600, 4640220

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The a(1) = 1 through a(6) = 13 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (o(oo))    (o(ooo))
                          (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   ((o(oo)))
                          ((((o))))  (o((oo)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The Matula-Goebel numbers of these trees are given by A324845.
Cf. A000081, A290689, A306844, A318185.
Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324843, A324846, A324847, A324848, A324849.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[!submultQ[b,#],{b,DeleteCases[#,{}]}]&];
Table[Length[rallt[n]],{n,10}]

A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 14, 23, 46, 85, 165, 313, 625, 1225, 2459, 4919, 9928, 20078, 40926, 83592

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

			The a(1) = 1 through a(7) = 14 fully recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((((oo))))   ((oo)(oo))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o))(oo))
                                                  (((o)(oo)))
                                                  ((o)((oo)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Gus Wiseman, The a(8) = 23 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(9) = 46 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(10) = 85 fully recursively anti-transitive rooted trees.

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324768, A324769, A324770.
Cf. A324838, A324841, A324844, A324846.

dallt[n_]:=Select[Union[Sort/@Join@@(Tuples[dallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&];
Table[Length[dallt[n]],{n,10}]

A324768 Number of fully anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 27, 60, 152, 376, 968, 2492, 6549, 17259, 46000, 123214, 332304, 900406, 2451999, 6703925

Offset: 1

Gus Wiseman, Mar 17 2019

An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.

			The a(1) = 1 through a(6) = 11 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (((((o)))))

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324769, A324770.
Cf. A324838, A324840, A324844, A324846.

rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&]],{n,10}]

a(17)-a(20) from Jinyuan Wang, Jun 20 2020

A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288

Offset: 1

Gus Wiseman, Mar 18 2019

			The a(1) = 1 through a(6) = 10 rooted trees:
  o  ((o))  ((oo))   ((ooo))    ((oooo))
            (((o)))  (((oo)))   (((ooo)))
                     ((o)(o))   ((o)(oo))
                     ((o(o)))   ((o(oo)))
                     ((((o))))  ((oo(o)))
                                ((((oo))))
                                (((o)(o)))
                                (((o(o))))
                                ((o((o))))
                                (((((o)))))

Cf. A000081, A290689, A304360, A306844, A317787, A318185.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324765, A324768, A324771, A324839, A324840, A324844, A324846.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],And@@Table[!submultQ[b,#],{b,#}]&]],{n,10}]

A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 15, 17, 31, 35, 57, 70, 111, 136, 213, 265, 405, 517, 763, 987, 1458, 1893, 2736, 3611, 5161, 6836, 9702

Offset: 1

Gus Wiseman, Mar 18 2019

A subset of totally transitive rooted trees (A318185).

			The a(1) = 1 through a(8) = 8 rooted trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
                (o(o))  (oo(o))  (oo(oo))   (ooo(oo))   (ooo(ooo))
                                 (ooo(o))   (oooo(o))   (oooo(oo))
                                 (o(o)(o))  (oo(o)(o))  (ooooo(o))
                                                        (oo(o)(oo))
                                                        (ooo(o)(o))
                                                        (o(o)(o)(o))
                                                        (o(o)(o(o)))

The Matula-Goebel numbers of these trees are given by A324842.
Cf. A000081, A279861, A290689, A290822, A318185.
Cf. A324704, A324736, A324748, A324753, A324847, A324848, A324854.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[submultQ[b,#],{b,#}]&];
Table[Length[rallt[n]],{n,10}]

A358453 Number of transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 37, 83, 190, 444, 1051, 2518, 6090, 14852

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root. An undirected version is A358454.

			The a(1) = 1 through a(7) = 17 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(o)o)  (o(o)oo)   (o(o)ooo)
                        (o(oo))  (o(oo)o)   (o(oo)oo)
                        (oo(o))  (o(ooo))   (o(ooo)o)
                                 (oo(o)o)   (o(oooo))
                                 (oo(oo))   (oo(o)oo)
                                 (ooo(o))   (oo(oo)o)
                                 (o(o)(o))  (oo(ooo))
                                            (ooo(o)o)
                                            (ooo(oo))
                                            (oooo(o))
                                            (o(o)(o)o)
                                            (o(o)(oo))
                                            (o(o)o(o))
                                            (o(oo)(o))
                                            (oo(o)(o))
                                            (o(o)((o)))

The unordered version is A290689, ranked by A290822.
The undirected version is A358454, ranked by A358458.
These trees are ranked by A358457.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]]]],{n,10}]

A143363 Number of ordered trees with n edges and having no protected vertices. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

Original entry on oeis.org

1, 1, 1, 3, 6, 17, 43, 123, 343, 1004, 2938, 8791, 26456, 80597, 247091, 763507, 2372334, 7413119, 23271657, 73376140, 232238350, 737638868, 2350318688, 7510620143, 24064672921, 77294975952, 248832007318, 802737926643

Offset: 0

Emeric Deutsch, Aug 20 2008

nonn

The "no protected vertices" condition can be rephrased as "every non-leaf vertex has at least one leaf child". But a(n) is also the number of ordered trees with n edges in which every non-leaf vertex has at most one leaf child. - David Callan, Aug 22 2014
Also the number of locally non-intersecting ordered rooted trees with n edges, meaning every non-leaf subtree has empty intersection. The unordered version is A007562. - Gus Wiseman, Nov 19 2022
a(n) is the number of parking functions of size n-1 avoiding the patterns 123, 132, and 213 . - Lara Pudwell, Apr 10 2023
For n>0, a(n) is the number of ways to place non-intersecting diagonals in convex n+3-gon so as to create no triangles such that none of the dividing diagonals passes through a chosen vertex. (empirical observation) - Muhammed Sefa Saydam, Feb 14 2025 and Aug 05 2025

			From _Gus Wiseman_, Nov 19 2022: (Start)
The a(0) = 1 through a(4) = 6 trees with at least one leaf directly under any non-leaf node:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                (o(o))  ((oo)o)
                        (o(o)o)
                        (o(oo))
                        (oo(o))
The a(0) = 1 through a(4) = 6 trees with at most one leaf directly under any node:
  o  (o)  ((o))  ((o)o)   (((o))o)
                 (o(o))   (((o)o))
                 (((o)))  ((o)(o))
                          ((o(o)))
                          (o((o)))
                          ((((o))))
(End)

Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Gi-Sang Cheon and Louis W. Shapiro, Protected points in ordered trees, Appl. Math. Letters, 21, 2008, 516-520.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Cf. A143362.
For exactly one leaf directly under any node we have A006013.
The unordered version is A007562, ranked by A316470.
Allowing lone children gives A319378.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
A358460 counts locally disjoint ordered trees, unordered A316473.
Cf. A318185, A324756, A324767, A324840, A324844.

p:=z^2*G^3-2*z*G^2-2*z^2*G^2+3*z*G+G+z^2*G-1-2*z=0: G:=RootOf(p,G): Gser:= series(G,z=0,33): seq(coeff(Gser,z,n),n=0..28);

a[n_Integer] := a[n] = Round[SeriesCoefficient[2 (x + 1 - Sqrt[x^2 - x + 1] Cos[ArcTan[(3 x Sqrt[12 x^3 - 96 x^2 - 24 x + 15])/(2 x^3 - 30 x^2 - 3 x + 2)]/3])/(3 x), {x, 0, n}]]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 10 2022 *)
RecurrenceTable[{25 (n + 5) (n + 6) a[n + 5] - 10 (n + 5) (5 n + 21) a[n + 4] - 2 (77 n^2 + 613 n + 1185) a[n + 3] + 2 (50 n^2 + 253 n + 312) a[n + 2] + 4 (2 n + 1) (7 n + 9) a[n + 1] - 4 n (2 n + 1) a[n] == 0, a[0] == 1, a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 6}, a[n], {n, 0, 27}] (* Vladimir Reshetnikov, Apr 11 2022 *)
ait[n_]:=ait[n]=If[n==1,{{}},Join@@Table[Select[Tuples[ait/@c],MemberQ[#,{}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[ait[n]],{n,15}] (* Gus Wiseman, Nov 19 2022 *)

a(n) = A143362(n,0) for n>=1.
G.f.: G=G(z) satisfies z^2*G^3-2z(1+z)G^2+(1+3z+z^2)G-(1+2z)=0.
G.f.: (x+1-sqrt(x^2-x+1)*cos(arctan((3*x*sqrt(12*x^3-96*x^2-24*x+15))/(2*x^3-30*x^2-3*x+2))/3))*2/(3*x). - Vladimir Reshetnikov, Apr 10 2022
Recurrence: 25*(n+5)*(n+6)*a(n+5) - 10*(n+5)*(5*n+21)*a(n+4) - 2*(77*n^2+613*n+1185)*a(n+3) + 2*(50*n^2+253*n+312)*a(n+2) + 4*(2*n+1)*(7*n+9)*a(n+1) - 4*n*(2*n+1)*a(n) = 0. - Vladimir Reshetnikov, Apr 11 2022
From Muhammed Sefa Saydam, Jul 12 2025: (Start)
a(n) =  Sum_{k=2..n+2} A046736(k) * A046736(n-k+3) , for n >= 0 and A046736(1) = 1.
a(n) = A049125(n) + Sum_{k=1..n-2} A049125(k) * A046736(n-k+2), for n >= 3.
a(n) = A049125(n) + Sum_{k=1..n-2} a(k) * a(n-k-1), for n >= 3. (End)

cp's OEIS Frontend

A290689 Number of transitive rooted trees with n nodes.

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A324764 Number of anti-transitive rooted identity trees with n nodes.

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A324765 Number of recursively anti-transitive rooted trees with n nodes.

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A324844 Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

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A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

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A324768 Number of fully anti-transitive rooted trees with n nodes.

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A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

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A324843 Number of unlabeled rooted trees with n nodes where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

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A358453 Number of transitive ordered rooted trees with n nodes.

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