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

Author

Gus Wiseman, Nov 29 2022

Keywords

• nonn

Comments

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

Examples

			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)))
		

Links

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

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.

Programs

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}]

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

Extensions

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