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

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

1,4

Comments

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

Links

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

Example

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)))

Mathematica

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}]

Prog

(PARI) \\ 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

Crossrefs

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.

Sequence in context: A172285 A392428 A192232 * A243320 A363470 A319897

Adjacent sequences: A358720 A358721 A358722 * A358724 A358725 A358726

Keyword

nonn

Author

Gus Wiseman, Nov 29 2022

Extensions

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

Status

approved