A048815 -id:A048815 - cp's OEIS frontend

A048816 Number of rooted trees with n nodes with every leaf at the same height.

1, 1, 2, 3, 5, 7, 12, 17, 28, 42, 68, 103, 168, 260, 420, 665, 1075, 1716, 2787, 4489, 7304, 11849, 19333, 31504, 51561, 84347, 138378, 227096, 373445, 614441, 1012583, 1669774, 2756951, 4555183, 7533988, 12469301, 20655523, 34238310, 56795325, 94270949

Offset: 1

Christian G. Bower, Apr 15 1999

nonn

The trees are unordered (see A000081). For balanced ordered rooted trees see A079500. - Joerg Arndt, Jul 20 2014

The trees are unlabeled. For labeled version see A238372. - Alois P. Heinz, Dec 29 2014

			See Arndt link.
From _Gus Wiseman_, Oct 08 2018: (Start)
The a(1) = 1 through a(7) = 12 balanced rooted trees with n nodes:
  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)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..3500 (terms 1..300 from Alois P. Heinz)
Joerg Arndt, balanced unordered rooted trees for n = 1..10
Index entries for sequences related to rooted trees

Cf. A048808-A048815.
Row sums of A244925.
Cf. A079500, A238372.
Cf. A000081, A000311, A001678, A120803, A320154, A320160, A316624, A320169.

T[n_, k_] := T[n, k] = If[n==1, 1, If[k==0, 0, Sum[Sum[If[dJean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 7, 7, 6, 3, 2, 1, 1, 0, 1, 8, 12, 8, 6, 3, 2, 1, 1, 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1, 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1, 0, 1, 21, 42, 39, 26, 17, 9, 6, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Jul 08 2014

			The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height 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  :
  :         :             :       :     :
  : ---1--- : -----2----- : --3-- : -4- :
Thus row 5 = [0, 1, 2, 1, 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  1,  1;
  0, 1,  2,  1,  1;
  0, 1,  2,  2,  1,  1;
  0, 1,  4,  3,  2,  1, 1;
  0, 1,  4,  5,  3,  2, 1, 1;
  0, 1,  7,  7,  6,  3, 2, 1, 1;
  0, 1,  8, 12,  8,  6, 3, 2, 1, 1;
  0, 1, 12, 18, 15,  9, 6, 3, 2, 1, 1;
  0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=0-10 give: A000007(n-1), A000012 (for n>0), A002865(n-1) (for n>2), A048808, A048809, A048810, A048811, A048812, A048813, A048814, A048815.
T(2n+1,n) gives A074045.
Row sums give A048816.

with(numtheory):
T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
      add(add(`if`(d

T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[dJean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

A074045 Number of rooted trees of 2n+1 nodes with every leaf at height n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 145, 246, 427, 729, 1256, 2152, 3702, 6341, 10892, 18662, 32016, 54853, 94034, 161055, 275929, 472461, 809033, 1384848, 2370434, 4056309, 6940744, 11873769, 20311018, 34737781, 59405959, 101577454, 173669088, 296890268

Offset: 0

Christian G. Bower, Aug 13 2002. Suggested by Paul D. Hanna

Vaclav Kotesovec, Table of n, a(n) for n = 0..3500 (terms 0..250 from Alois P. Heinz)
Index entries for sequences related to rooted trees

Limit of A048808-A048815.

Cf. A244925.

a(n) = A244925(2n+1,n).

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A048816 Number of rooted trees with n nodes with every leaf at the same height.

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A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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A074045 Number of rooted trees of 2n+1 nodes with every leaf at height n.

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