Mareike Fischer - cp's OEIS frontend

A386912 The base-2 logarithm of this sequence equals the Q statistic value of the greedy-from-the-bottom tree with n leaves, which is the minimal Q statistic value for n.

1, 2, 6, 16, 60, 192, 672, 2048, 8640, 30720, 118272, 393216, 1597440, 5505024, 20643840, 67108864, 300810240, 1132462080, 4602200064, 16106127360, 68702699520, 248034361344, 972407439360, 3298534883328, 14495514624000, 53601191854080

Offset: 1

Mareike Fischer, Aug 07 2025

nonn

			a(1)=1, because a(1)=Product(i=2..1)=1 (empty product).
a(2)=2, because q(2,2)=1, so a(2)=2^q(2,2)=2^1=2.
a(3)=6, because q(3,2)=1 and q(3,3)=1, so a(3)=2^q(3,2)*3^q(3,3)=2^1*3^1=6.

Alois P. Heinz, Table of n, a(n) for n = 1..1666 (first 300 terms from Mareike Fischer)
Sean Cleary, Mareike Fischer, and Katherine St. John, The GFB Tree and Tree Imbalance Indices, arXiv:2502.12854 [q-bio.PE], 2025.

Cf. A132862.

a:= proc(n) option remember; `if`(n<2, 1, (b-> (f->
      a(f)*n*a(n-f))(min(b, n-b/2)))(2^ilog2(n-1)))
    end:
seq(a(n), n=1..26);  # Alois P. Heinz, Aug 18 2025

q[n_, i_] := Module[{factor, ki}, factor = FactorInteger[i];
  ki = Ceiling[Log2[i]];
  If[(i == 1 || i == n), Return[n/i],
   If[(Length[factor] == 1 && factor[[1]][[1]] == 2),
     If[(Mod[n, i] == 0 || Mod[n, i] >= 2^(ki - 1)),
      Return[Floor[n/i]], Return[Floor[n/i] - 1]],
     If[Mod[n - i, 2^(ki - 1)] == 0, Return[1], Return[0]]];
   ]];
a[n_] := Product[i^q[n, i], {i, 2, n}]; alist = {};
For[n = 1, n <= 300, n++, AppendTo[alist, a[n]]]; Print[alist]
(* Alternative, after Alois P. Heinz: *)
A386912[n_] := A386912[n] = If[n == 1, 1, n*A386912[#]*A386912[n - #]] & [Min[#, n - #/2] & [2^(BitLength[n - 1] - 1)]]; Array[A386912, 30] (* Paolo Xausa, Aug 19 2025 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n: int) -> int:
    if n < 2: return 1
    b = 1 << ((n - 1).bit_length() - 1)
    f = min(b, n - b // 2)
    return a(f) * n * a(n - f)
print([a(n) for n in range(1, 27)])  # after Alois P. Heinz, Peter Luschny, Aug 19 2025

a(n) = Product_{i=2..n} i^q(n,i), with k(i)=ceiling(log_2(i)) and q(n,i)=floor(n/i) if (i=2^(k(i)) and ((n mod i)=0 or (n mod i)>=2^(k(i)-1))), and q(n,i)=floor(n/i)-1 if (i=2^(k(i)) and (0 < (n mod i)< 2^(k(i)-1))) and q(n,i)=1 if (i!=2^(k(i)) and ( (n-i) mod 2^(k(i)-1)) =0 ) and q(n,i)=0 if (i!=2^(k(i)) and ( (n-i) mod 2^(k(i)-1)) >0 ).
a(n) = Product_{i=0..k(n)-1} (2^(k(n)-i))^(2^i) if d(n)=2^(k(n)-1). Else, a(n) = Product_{i=0..k(n)-2} (2^(k(n)-i-1))^(2^i) * 2^d(n) * Product_{i=1..k(n)-1} ( 2^floor(d(n)/(2^i)) * ((2^i + (d(n)-2^i*(floor(d(n)/(2^i))))) / (2^i) )^(ceiling(d(n)/2^i)-floor(d(n)/(2^i))) if d(n) < 2^(k(n)-1), with k(n)=ceiling(log_2(n)) and d(n)=n-2^(k(n)-1).
a(n) = n*a(f)*a(n-f) with f = min(b,n-b/2) and b = 2^floor(log_2(n-1)) for n>1, a(1) = 1. - Alois P. Heinz, Aug 18 2025

A344852 Number of rooted binary trees with n leaves with minimal Symmetry Nodes Index (SNI) or, equivalently, with the maximal number of symmetry nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 3, 15, 1, 1, 1, 3, 1, 3, 3, 15, 1, 3, 3, 15, 3, 15, 15, 105, 1, 1, 1, 3, 1, 3, 3, 15, 1, 3, 3, 15, 3, 15, 15, 105, 1, 3, 3, 15, 3, 15, 15, 105, 3, 15, 15, 105, 15, 105, 105, 945, 1, 1, 1, 3, 1, 3, 3, 15, 1, 3, 3, 15, 3, 15, 15, 105, 1, 3, 3, 15, 3, 15, 15, 105, 3, 15, 15, 105, 15, 105, 105, 945, 1

Offset: 1

Mareike Fischer, Jun 15 2021

nonn

In a rooted binary tree, each internal node has precisely two children. An internal node is called a symmetry node if the subtrees of its two children are isomorphic. The Symmetry Nodes Index SNI, however, counts the internal nodes which are not symmetry nodes. The minimal SNI of a rooted binary tree with n leaves is wt(n)-1, where wt(n) = A000120(n) denotes the binary weight of n, i.e., it refers to the number of 1's in the binary expansion of n.

a(n) is also the number of trees with n leaves and minimal Rogers's J tree balance index, which is the number of internal nodes which are not balanced, i.e., the number of nodes whose left and right subtrees do not have the same number of leaves.

			a(7)=2, because both trees (((x,x),(x,x)),((x,x),x)) and (((x,x),(x,x),x),(x,x)) have four inner nodes which are symmetry nodes and only two inner nodes which are non-symmetry nodes. So the SNI of both trees equals 2, which is minimal for n=7.
a(8)=1, because only the tree (((x,x),(x,x)),((x,x),(x,x))) is minimal: its SNI equals 0.

Alois P. Heinz, Table of n, a(n) for n = 1..16383
S. J. Kersting and M. Fischer, Measuring tree balance using symmetry nodes - a new balance index and its extremal properties, arXiv:2105.00719 [q-bio.PE], 2021.

Cf. A000120, A001147.

a:= n-> doublefactorial(2*add(i, i=convert(n, base, 2))-3):
seq(a(n), n=1..100); # Georg Fischer, Jun 15 2021

a[n_] := (2*DigitCount[n, 2, 1] - 3)!!; Table[a[n], {n, 1, 100}]

a(n)={my(t=hammingweight(n)-1); (2*t)!/(2^t*t!)} \\ Andrew Howroyd, Jun 15 2021

a(n) = (2*wt(n) - 3)!! = A001147(2*A000120(n) - 3).

A345135 Number of ordered rooted binary trees with n leaves and with minimal Sackin tree balance index.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 6, 4, 1, 8, 28, 56, 70, 56, 28, 8, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 32, 496, 4960, 35960, 201376, 906192, 3365856, 10518300, 28048800, 64512240, 129024480, 225792840, 347373600, 471435600

Offset: 0

Mareike Fischer, Jun 09 2021

Ordered rooted binary trees are trees with two descendants per inner node where left and right are distinguished.
The Sackin tree balance index is also known as total external path length.
a(0) = 1 by convention.

			a(1) = a(2) = 1 because there are only the trees (o) and (o,o) which get counted. a(3) = 2 because the trees ((o,o),o) and (o,(o,o)) get counted. a(4) = 1 because only the tree ((o,o),(o,o)) is counted. Note that the other possible rooted binary ordered trees with four leaves, namely the different orderings of (((o,o),o),o), are not Sackin minimal. a(5) = 4 because the following trees get counted: (((o,o),o),(o,o)), ((o,(o,o)),(o,o)), ((o,o),((o,o),o)), ((o,o),(o,(o,o))).

Alois P. Heinz, Table of n, a(n) for n = 0..4096
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index. Ann. Comb. 25, 515-541 (2021), Theorem 7.

Cf. A000079, A000108, A011782, A037293, A051179, A053644, A299037.

a:= n-> (b-> binomial(b, n-b))(2^ilog2(n)):
seq(a(n), n=0..46);  # Alois P. Heinz, Jun 09 2021

a[0] := 1; a[n_] := Module[{k = 2^(BitLength[n] - 1)}, Binomial[k, n - k]];
Table[a[n], {n, 0, 46}]

a(n) = binomial(2^(ceiling(log_2(n))-1),n-2^(ceiling(log_2(n))-1)).
a(n) = binomial(A053644(n),n-A053644(n)).
a(2^n) = 1.
a(2^n-1) = A011782(n).
a(2^n+1) = A000079(n).
From Alois P. Heinz, Jun 09 2021: (Start)
max({ a(k) | k = 2^n..2^(n+1) }) = A037293(n).
Sum_{i=2^n..2^(n+1)-1} a(i) = 2^(2^n) - 1 = A051179(n). (End)
Conjecture: Sum_{n>=0} a(n)*x^n = 1 + x + Sum_{n>=0} x^(2^n) * ((1 + x)^(2^n) - 1). - Paul D. Hanna, Jul 18 2024

A344934 Number of rooted binary phylogenetic trees with n leaves and minimal Sackin tree balance index.

Original entry on oeis.org

1, 1, 3, 3, 30, 135, 315, 315, 11340, 198450, 2182950, 16372125, 85135050, 297972675, 638512875, 638512875, 86837751000, 5861548192500, 259861969867500, 8445514020693750, 212826953321482500, 4292010225316563750, 70511596558772118750, 951906553543423603125, 10576739483815817812500, 96248329302723942093750

Offset: 1

Mareike Fischer, Jun 09 2021

nonn

Rooted binary phylogenetic trees with n leaves are rooted trees for which each internal node has precisely two children and whose leaves are bijectively labeled by the set {1,...,n}.

Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. 25, 515-541 (2021).

Cf. A299037, A345135.

a[n_] := Module[{k = Ceiling[Log2[n]], int, na, nb, sum, i},
  If[n == 1, Return[1],
   int = IntegerPartitions[n, {2}];
   If[OddQ[n], sum = 0, sum = 1/2*Binomial[n, n/2]*((a[n/2])^2)];
   For[i = 1, i <= Length[int], i++,
    na = int[[i]][[1]]; nb = int[[i]][[2]];
    If[na > n/2 && na <= 2^(k - 1) && nb >= 2^(k - 2),
     sum = sum + Binomial[n, na]*a[na]*a[nb];
     ];
    ];
   Return[sum];
   ]]

seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, my(k=1+logint(n-1,2)); a[n]=if(n%2==0, a[n/2]*binomial(n,n/2)/2) + sum(i=n\2+1, min(2^(k-1), n-2^(k-2)), binomial(n,i)*a[i]*a[n-i])); a} \\ Andrew Howroyd, Jun 09 2021

With k:=log_2(n) and g(n):=0 if n is odd and g(n) := (1/2)*binomial(n,n/2)*a(n/2) if n is even and pairs := set of all pairs (na,nb) such that na+nb=n and na >= nb and na > n/2 and na <= 2^(k-1) and nb >= 2^(k-2), we get:
a(n) = g(n) + sum over all described pairs (na,nb): binomial(n,na)*a(na)*a(nb).
a(n) = g(n) + Sum_{i=floor(n/2)+1..2^(k-1), i <= 2^(k-2)} binomial(n,i)*a(i)*a(n-i), where k = ceiling(log_2(n)) and g(n)=0 for odd n, g(n) = binomial(n,n/2)*a(n/2)/2 for even n.

A344613 Number of rooted binary trees (in which all inner nodes have precisely two children) with n leaves and with maximal number of cherries (i.e., maximal number of pendant subtrees with two leaves).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 2, 9, 3, 20, 6, 46, 11, 106, 23, 248, 46, 582, 98, 1376, 207, 3264, 451, 7777, 983, 18581, 2179, 44526, 4850, 106936, 10905, 257379, 24631, 620577, 56011, 1498788, 127912, 3625026, 293547, 8779271, 676157, 21287278, 1563372, 51671864, 3626149, 125550018

Offset: 1

Mareike Fischer, Jun 09 2021

nonn

This sequence describes the number of rooted binary trees with n leaves with maximal cherry tree balance index or, equivalently, with minimal modified cherry tree balance index.

Alois P. Heinz, Table of n, a(n) for n = 1..5000
S. J. Kersting and M. Fischer, Measuring tree balance using symmetry nodes - a new balance index and its extremal properties, arXiv:2105.00719 [q-bio.PE], 2021.

Cf. A001190, A085748.

b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
    end:
g:= proc(n) option remember; `if`(n<1, 1,
      add(g(n-i)*b(i), i=1..n))
    end:
a:= n-> `if`(n::even, b(n/2), g((n-1)/2)):
seq(a(n), n=1..52);  # Alois P. Heinz, Jun 09 2021

(* WE generates the Wedderburn Etherington Numbers, OEIS sequence A001190 *)
WE[n_] := Module[{i},
   If[n == 1, Return[1],
    If[Mod[n, 2] == 0,
     Return[
      WE[n/2]*(WE[n/2] + 1)/2 + Sum[WE[i]*WE[n - i], {i, 1, n/2 - 1}]],
     Return[Sum[WE[i]*WE[n - i], {i, 1, Floor[n/2]}]]
     ]
    ]
   ];
(* b is just a support function *)
b[n_] := b[n] =
   If[n < 2, n,
    If[OddQ[n], 0, Function[t, t*(1 - t)/2][b[n/2]]] +
     Sum[b[i]*b[n - i], {i, 1, n/2}]];
(* c generates the elements of OEIS sequence A085748 *)
c[n_] := c[n] = If[n < 1, 1, Sum[c[n - i]*b[i], {i, 1, n}]];
(* a generates the number of rooted binary trees with maximal number of cherries *)
a[n_] := Module[{},
  If[EvenQ[n], Return[WE[n/2]], Return[c[(n - 1)/2]]]]

a(n) = A001190(n/2) if n even, otherwise a(n) = A085748((n-1)/2).

A307689 The number of rooted binary trees on n leaves with minimal Colless index.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 3, 3, 1, 1, 1, 4, 6, 10, 16, 21, 13, 11, 13, 21, 16, 10, 6, 4, 1, 1, 1, 5, 10, 20, 50, 82, 73, 66, 184, 411, 548, 351, 455, 326, 144, 67, 144, 326, 455, 351, 548, 411, 184, 66, 73, 82, 50, 20, 10, 5, 1, 1

Offset: 1

Mareike Fischer, Apr 22 2019

a(n) is the number of maximally balanced binary rooted trees with n leaves according to the Colless imbalance index. It is bounded above by sequence A299037.

			There are 13 trees with minimal Colless index and 23 leaves, i.e. a(23)=13.

Mareike Fischer, Table of n, a(n) for n = 1..512
Tomás M. Coronado and Francesc Rosselló, The minimum value of the Colless index arXiv:1903.11670 [q-bio.PE], 2019.
Mareike Fischer, Lina Herbst, and Kristina Wicke, Extremal properties of the Colless tree balance index for rooted binary trees, arXiv:1904.09771 [math.CO], 2019.
Tree Balance, Colless index

The sequence is bounded above by A299037.

The sequence of the number of trees with minimal Colless index is also linked to the values of the minimal Colless index as given by A296062.

(*QB[n] is just a support function -- the function that actually generates the sequence of the numbers of trees with minimal Colless index and n leaves is given by minCbasedonQB; see below*)
QB[n_] := Module[{k, n0, bin, l, m = {}, i, length, qb = {}, j},
  If[OddQ[n], k = 0,
   k = FactorInteger[n][[1]][[2]];
   ];
  n0 = n/(2^k);
  bin = IntegerDigits[n0, 2];
  length = Length[bin];
  For[i = Length[bin], i >= 1, i--,
   If[bin[[i]] != 0, PrependTo[m, length - i]];
   ];
  l = Length[m];
  If[l == 1, Return[{{n/2, n/2}}],
   AppendTo[
    qb, {2^k*(Sum[2^(m[[i]] - 1), {i, 1, l - 1}] + 1),
     2^k*(Sum[2^(m[[i]] - 1), {i, 1, l - 1}])}];
   For[j = 2, j <= l - 1, j++,
    If[m[[j]] > m[[j + 1]] + 1,
     AppendTo[
      qb, {2^k*(Sum[2^(m[[i]] - 1), {i, 1, j - 1}] + 2^m[[j]]),
       n - 2^k*(Sum[2^(m[[i]] - 1), {i, 1, j - 1}] + 2^m[[j]])}]];
    If[m[[j]] < m[[j - 1]] - 1,
     AppendTo[
      qb, {n - 2^k*Sum[2^(m[[i]] - 1), {i, 1, j - 1}],
       2^k*Sum[2^(m[[i]] - 1), {i, 1, j - 1}]}]];
    ];
   If[k >= 1, AppendTo[qb, {n/2, n/2}]];
   Return[qb];
   ];
  ]
minCbasedonQB[n_] := Module[{qb = QB[n], min = 0, i, na, nb},
  If[n == 1, Return[1],
   For[i = 1, i <= Length[qb], i++,
    na = qb[[i]][[1]]; nb = qb[[i]][[2]];
    If[na != nb, min = min + minCbasedonQB[na]*minCbasedonQB[nb],
     min = min + Binomial[minCbasedonQB[n/2] + 1, 2]];
    ];
   Return[min];
   ]
  ]

a(1)=1; a(n) = Sum_{(n_a,n_b): n_a+n_b=n, n_a > n_b, (n_a,n_b) in QB(n)}} ( a(n_a)* a(n_b)) +f(n), where f(n)=0 if n is odd} and f(n)= binomial(a(n/2)+1,2) if n is even; and where QB(n)={(n_a,n_b): n_a+n_b=n and such that there exists a tree T on n leaves with minimal Colless index and with leaf partitioning (n_a,n_b)} }.

A299037 a(n) is the number of rooted binary trees with minimal Sackin index and n leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 5, 3, 3, 1, 1, 1, 4, 6, 14, 17, 27, 28, 35, 28, 27, 17, 14, 6, 4, 1, 1, 1, 5, 10, 30, 55, 121, 207, 378, 575, 894, 1217, 1651, 1993, 2373, 2546, 2682, 2546, 2373, 1993, 1651, 1217, 894, 575, 378, 207, 121, 55, 30, 10, 5, 1, 1, 1, 6, 15, 55, 135, 381, 903, 2205, 4848, 10599, 21631

Offset: 1

Mareike Fischer, Feb 01 2018

a(n) is also the number of maximally balanced trees with n leaves according to the Sackin index.
From Omar E. Pol, Feb 02 2018: (Start)
Also the sequence can be written as an irregular triangle read by rows in which the row lengths are the terms of A011782 (see example section).
Column 1 gives A000012. Column 2 gives A000027.
Conjecture 1: column 3 gives the nonzero terms of A000217.
Conjecture 2: column 4 gives the nonzero terms of A000330. (End)
Comment from the author, Feb 02 2018: Denote the (i,j)-th entry of the irregular triangle described above (row i, column j) with T(i,j), i >= 1 and 1 <= j <= A011782(i-1). Then, rows 1 and 2 contain a 1 each (and they denote the number of trees minimizing the Sackin index with 1 and two leaves, respectively) and row i ranges from 2^(i-2) + 1 to 2^(i-1) leaves and for each of these numbers gives the number of trees with minimum Sackin index. E.g., row 4 gives the number of such trees for 2^(4-2) + 1 = 5 leaves up to 2^(4-1) = 8 leaves.
From Omar E. Pol, Mar 01 2018: (Start)
Conjecture 3: column 5 gives the nonzero terms of A212415.
Conjecture 4: row sums give 1 together with A006893.
Conjecture 5: T(i,j) has i >= 0 where "i" is the height of the rooted trees.
Conjecture 6: For i >= 1, j is the number of leaves minus 2^(i-1). (End)

			Whenever n = 2^k, i.e., n is a power of 2, then the tree minimizing the Sackin index is unique, namely the so-called fully balanced tree of height k.
From _Omar E. Pol_, Feb 01 2018: (Start)
Written as an irregular triangle the sequence begins:
  1;
  1;
  1, 1;
  1, 2, 1,  1;
  1, 3, 3,  5,  3,  3,  1,  1;
  1, 4, 6, 14, 17, 27, 28, 35, 28, 27, 17, 14, 6, 4, 1, 1;
  ... (End)

Mareike Fischer, Table of n, a(n) for n = 1..1024
Mareike Fischer, Extremal values of the Sackin balance index for rooted binary trees, arXiv:1801.10418 [q-bio.PE], 2018, (see theorem 8 and subsequent remark).
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 5.
Index entries for sequences related to rooted trees

Cf. A001190 (number of rooted binary trees with n leaves), A011782.

maxnumber = 1024;
minlist = {1}; For[n = 2, n <= maxnumber, n++,
parts = IntegerPartitions[n, {2}];
total = 0;
For[s = 1, s <= Length[parts], s++,
  na = parts[[s]][[1]]; nb = parts[[s]][[2]]; k = Ceiling[Log2[n]];
  ka = Ceiling[Log2[na]]; kb = Ceiling[Log2[nb]];
  If[na >= n/2 && ka == k - 1 && kb == k - 1 && na != nb,
   total = total + minlist[[na]]*minlist[[nb]],
   If[na >= n/2 && ka == k - 1 && kb == k - 1 && na == nb,
    total = total + Binomial[minlist[[na]] - 1 + 2, 2],
    If[na >= n/2 && nb == 2^(k - 2) && ka == k - 1,
     total = total + minlist[[na]]]]];
  ]; AppendTo[minlist, total];
]

a(1) = 1.

a(n) = Sum_{(n_a,n_b): n_a+n_b=n, n_a>=n/2, ceiling(log_2(n_a)) = ceiling(log_2(n))-1, ceiling(log_2(n_b)) = ceiling(log_2(n))-1,n_a!=n_b} a(n_a)*a(n_b) + Sum_{(n_a,n_b): n_a+n_b=n, ceiling(log_2(n_a)) = ceiling(log_2(n)) - 1, ceiling(log_2(n_b)) = ceiling(log_2(n)) - 1, n_a = n_b} binomial(a(n_a)+1,2) + Sum_{(n_a,n_b): n_a+n_b=n, ceiling(log_2(n_a)) = ceiling(log_2(n)) - 1, n_b = 2^(ceiling(log_2(n))-2)} a(n_a).

cp's OEIS Frontend

User: Mareike Fischer

A386912 The base-2 logarithm of this sequence equals the Q statistic value of the greedy-from-the-bottom tree with n leaves, which is the minimal Q statistic value for n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A344852 Number of rooted binary trees with n leaves with minimal Symmetry Nodes Index (SNI) or, equivalently, with the maximal number of symmetry nodes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A345135 Number of ordered rooted binary trees with n leaves and with minimal Sackin tree balance index.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A344934 Number of rooted binary phylogenetic trees with n leaves and minimal Sackin tree balance index.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A344613 Number of rooted binary trees (in which all inner nodes have precisely two children) with n leaves and with maximal number of cherries (i.e., maximal number of pendant subtrees with two leaves).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A307689 The number of rooted binary trees on n leaves with minimal Colless index.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A299037 a(n) is the number of rooted binary trees with minimal Sackin index and n leaves.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs