A185650 -id:A185650 - cp's OEIS frontend

A358583 Number of rooted trees with n nodes, at least half of which are leaves.

1, 1, 1, 3, 4, 13, 20, 67, 110, 383, 663, 2346, 4217, 15118, 27979, 101092, 191440, 695474, 1341974, 4893067, 9589567, 35055011, 69612556, 254923825, 511987473, 1877232869, 3807503552, 13972144807, 28585315026, 104955228432, 216381073935, 794739865822

Offset: 1

Gus Wiseman, Nov 23 2022

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..200

For equality we have A185650 aerated, ranked by A358578.
The strict case is A358581.
The opposite version is A358584, strict A358582.
The ordered version is A358586, strict A358585.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square rooted trees, ranked by A358577, ordered A358590.
Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358586.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>=Count[#,[_],{0,Infinity}]&]],{n,1,10}]

\\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[(n-1)\2+1..#u]))} \\ Andrew Howroyd, Dec 31 2022

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=floor((n-1)/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022

A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 0, 1, 2, 1, 0, 0, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 0, 2, 1, 0, 0, 2, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 3, 0, 1, 1, 2, 0, 1

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The tree (o(o)((o))(oo)) with Matula-Goebel number 210 has edge-height 3 and 5 internal nodes, so a(210) = 2.

Positions of 0's are A209638, complement A358725.
Positions of 1's are A358576, counted by A358587.
Other differences: A358580, A358726, A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A316321, A358577, A358578, A358592, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],[_],{0,Infinity}]-(Depth[MGTree[n]]-2),{n,100}]

a(n) = A342507(n) - A109082(n).

A358726 Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 1, -1, 1, 2, 4, 0, 2, 0, 2, -2, 2, 0, 0, 1, 0, 3, 2, -1, 2, 1, 0, -1, 3, 1, 5, -3, 3, 1, 1, -1, 1, -1, 1, 0, 3, -1, 1, 2, 1, 1, 3, -2, -1, 1, 1, 0, -1, -1, 3, -2, -1, 2, 3, 0, 1, 4, -1, -4, 1, 2, 1, 0, 1, 0, 2, -2, 1, 0, 1, -2, 2, 0, 4, -1

Offset: 1

Gus Wiseman, Nov 29 2022

sign

Node-height is the number of nodes in the longest path from root to leaf.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The tree (oo(oo(o))) with Matula-Goebel number 148 has node-height 4 and 5 leaves, so a(148) = -1.

Positions of first appearances are A007097 and latter terms of A000079.
Positions of 0's are A358577.
Other differences: A358580, A358724, A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A056239, A112798.
Cf. A185650, A206487, A209638, A358576, A358578, A358587, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[(Depth[MGTree[n]]-1)-Count[MGTree[n],{},{0,Infinity}],{n,1000}]

a(n) = A358552(n) - A109129(n).

A358725 Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.

Original entry on oeis.org

9, 15, 18, 21, 23, 25, 27, 30, 33, 35, 36, 39, 42, 45, 46, 47, 49, 50, 51, 54, 55, 57, 60, 61, 63, 65, 66, 69, 70, 72, 73, 75, 77, 78, 81, 83, 84, 85, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 108, 110, 111, 113, 114, 115, 117, 119, 120, 121

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The terms together with their corresponding trees begin:
   9: ((o)(o))
  15: ((o)((o)))
  18: (o(o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  30: (o(o)((o)))
  33: ((o)(((o))))
  35: (((o))(oo))
  36: (oo(o)(o))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  45: ((o)(o)((o)))
  46: (o((o)(o)))
  47: (((o)((o))))
  49: ((oo)(oo))
  50: (o((o))((o)))

Gus Wiseman, The first 64 ordered trees with a greater number of internal vertices than edge-height.

Complement of A209638 (the case of equality).
These trees are counted by A316321.
Positions of positive terms in A358724.
The case of equality for node-height is A358576.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936
A055277 counts rooted trees by nodes and leaves, ordered A001263.
Differences: A358580, A358724, A358726, A358729.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A358577, A358578, A358581-A358586, A358587, A358592, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[MGTree[#],[_],{0,Infinity}]>Depth[MGTree[#]]-2&]

A342507(a(n)) > A109082(a(n)).

A358107 Number of unlabeled trees covering 2n nodes, n+1 of which are leaves.

Original entry on oeis.org

1, 1, 2, 6, 26, 119, 626, 3495, 20688, 127339, 810418, 5293790, 35351571, 240478715, 1662071181, 11646620758, 82601643511, 592110678762, 4284830131865, 31271691087861, 229980550743717, 1703097703162249, 12691879796699486, 95129358337729084, 716801612475691847

Offset: 1

Gus Wiseman, Dec 02 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Gus Wiseman, The a(4) = 6 trees covering 8 nodes, 5 of which are leaves.

Central column of A055290.
The labeled version is the central column of A055314.
For n leaves we have A359398.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unordered A001433.
Cf. A185650, A358732.

Terms a(11) and beyond from Andrew Howroyd, Jan 01 2023

A358728 Number of n-node rooted trees whose node-height is less than their number of leaves.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 10, 30, 76, 219, 582, 1662, 4614, 13080, 36903, 105098, 298689, 852734, 2434660, 6964349, 19931147, 57100177, 163647811, 469290004, 1346225668, 3863239150, 11089085961, 31838349956, 91430943515, 262615909503, 754439588007, 2167711283560

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Node-height is the number of nodes in the longest path from root to leaf.

			The a(1) = 0 through a(7) = 10 trees:
  .  .  .  (ooo)  (oooo)  (ooooo)   (oooooo)
                          ((oooo))  ((ooooo))
                          (o(ooo))  (o(oooo))
                          (oo(oo))  (oo(ooo))
                          (ooo(o))  (ooo(oo))
                                    (oooo(o))
                                    ((o)(ooo))
                                    ((oo)(oo))
                                    (o(o)(oo))
                                    (oo(o)(o))

Andrew Howroyd, Table of n, a(n) for n = 1..200

These trees are ranked by A358727.
For internals instead of node-height we have A358581, ordered A358585.
The case of equality is A358589 (square trees), ranked by A358577.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
Cf. A109082, A109129, A185650, A358552, A358582-A358586, A358587, A358591.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Depth[#]-1

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->sum(j=h+1, n-1, polcoef(p,j,y))), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 12, 720, 109200, 31752000, 15186346560, 10852244282880, 10851787634688000, 14481281691676800000, 24881574582258352358400, 53525038934303849706393600, 140958354488116955062668595200, 446153762528143389466306560000000, 1671353230826683972965623004979200000

Offset: 1

Gus Wiseman, Dec 01 2022

nonn

			The a(2) = 12 trees:
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{3,4}}
  {{1,2},{1,4},{2,3}}
  {{1,2},{1,4},{3,4}}
  {{1,2},{2,3},{3,4}}
  {{1,2},{2,4},{3,4}}
  {{1,3},{1,4},{2,3}}
  {{1,3},{1,4},{2,4}}
  {{1,3},{2,3},{2,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4}}
  {{1,4},{2,3},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 1..100

A central column of A055314.
The unlabeled rooted version is A185650.
The unlabeled version is A358107.
A000272 counts trees, bisection A163395.
A001187 counts connected graphs.
A006129 counts covering graphs.
A014068 counts graphs with n vertices and n-1 edges.
Cf. A000055, A001349, A006125.

a[n_]:=StirlingS2[2*n-2, n]*(2*n)!/n!; Array[a,14] (* Stefano Spezia, Aug 02 2024 *)

a(n) = stirling(2*n-2, n, 2)*(2*n)!/n! \\ Andrew Howroyd, Dec 30 2022

a(n) = A055314(2*n, n) = Stirling2(2*n-2, n)*(2*n)!/n!. - Andrew Howroyd, Dec 30 2022

Terms a(6) and beyond from Andrew Howroyd, Dec 30 2022

A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 1, 2, 8, 32, 158, 833, 4755, 28389, 176542, 1131055, 7432876, 49873477, 340658595, 2362652648, 16605707901, 118082160358, 848399575321, 6152038125538, 44981009272740, 331344933928536, 2457372361637286, 18337490246234464, 137612955519565773, 1038076541372187991

Offset: 1

Gus Wiseman, Jan 01 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Gus Wiseman, The a(4) = 8 trees covering 8 nodes, half of which are leaves.

Left of central column of A055290.
The labeled version is the left of central column of A055314.
The rooted version is A185650.
For n+1 leaves we have A358107.
The labeled version is A358732.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unlabeled A001433.

a(n) = A055290(2*n, n). - Andrew Howroyd, Jan 01 2023

Terms a(12) and beyond from Andrew Howroyd, Jan 01 2023

A358727 Matula-Goebel numbers of rooted trees with greater number of leaves (width) than node-height.

Original entry on oeis.org

8, 16, 24, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 80, 81, 84, 96, 98, 104, 106, 108, 112, 114, 120, 126, 128, 131, 133, 136, 140, 144, 147, 148, 152, 156, 159, 160, 162, 168, 171, 172, 178, 180, 182, 184, 189, 190, 192, 196, 200, 204, 208

Offset: 1

Gus Wiseman, Dec 01 2022

nonn

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Node-height is the number of nodes in the longest path from root to leaf.

			The terms together with their corresponding rooted trees begin:
   8: (ooo)
  16: (oooo)
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  49: ((oo)(oo))
  53: ((oooo))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))

Gus Wiseman, The first 64 rooted trees with greater width than height.

Positions of negative terms in A358726.
These trees are counted by A358728.
Differences: A358580, A358724, A358726, A358729.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
MG core: A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A185650, A206487, A209638, A358576, A358577, A358578, A358587, A358730.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Depth[MGTree[#]]-1

A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

Original entry on oeis.org

0, 1, 0, 2, 1, 6, 7, 26, 43, 135, 276, 755, 1769, 4648, 11406, 29762, 75284, 195566, 503165, 1310705, 3402317, 8892807, 23231037, 60906456, 159786040, 420144405, 1105673058, 2914252306, 7688019511, 20304253421, 53667498236, 141976081288, 375858854594, 995728192169

Offset: 1

Gus Wiseman, Nov 29 2022

nonn

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

			The a(1) = 0 through a(7) = 7 trees:
  .  (o)  .  ((oo))  ((o)(o))  (((ooo)))  (((o))(oo))
             (o(o))            ((o(oo)))  (((o)(oo)))
                               ((oo(o)))  ((o)((oo)))
                               (o((oo)))  ((o)(o(o)))
                               (o(o(o)))  ((o(o)(o)))
                               (oo((o)))  (o((o)(o)))
                                          (o(o)((o)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

For internals instead of leaves: A011782, ranked by A209638.
For internals instead of edge-height: A185650 aerated, ranked by A358578.
For node-height: A358589 (square trees), ranked by A358577, ordered A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.
Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.

art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==Depth[#]-2&]],{n,1,10}]

\\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023

cp's OEIS Frontend

A358583 Number of rooted trees with n nodes, at least half of which are leaves.

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A358724 Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

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A358726 Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.

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A358725 Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.

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A358107 Number of unlabeled trees covering 2n nodes, n+1 of which are leaves.

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A358728 Number of n-node rooted trees whose node-height is less than their number of leaves.

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A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

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A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

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