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Number of n-node rooted trees of edge-height equal to their number of leaves.
1

%I #10 Jan 01 2023 14:46:05

%S 0,1,0,2,1,6,7,26,43,135,276,755,1769,4648,11406,29762,75284,195566,

%T 503165,1310705,3402317,8892807,23231037,60906456,159786040,420144405,

%U 1105673058,2914252306,7688019511,20304253421,53667498236,141976081288,375858854594,995728192169

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

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

%H Andrew Howroyd, <a href="/A358723/b358723.txt">Table of n, a(n) for n = 1..200</a>

%e The a(1) = 0 through a(7) = 7 trees:

%e . (o) . ((oo)) ((o)(o)) (((ooo))) (((o))(oo))

%e (o(o)) ((o(oo))) (((o)(oo)))

%e ((oo(o))) ((o)((oo)))

%e (o((oo))) ((o)(o(o)))

%e (o(o(o))) ((o(o)(o)))

%e (oo((o))) (o((o)(o)))

%e (o(o)((o)))

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

%t Table[Length[Select[art[n],Count[#,{},{-2}]==Depth[#]-2&]],{n,1,10}]

%o (PARI) \\ Needs R(n,f) defined in A358589.

%o seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ _Andrew Howroyd_, Jan 01 2023

%Y For internals instead of leaves: A011782, ranked by A209638.

%Y For internals instead of edge-height: A185650 aerated, ranked by A358578.

%Y For node-height: A358589 (square trees), ranked by A358577, ordered A358590.

%Y A000081 counts rooted trees, ordered A000108.

%Y A034781 counts rooted trees by nodes and height, ordered A080936.

%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.

%Y A358575 counts rooted trees by nodes and internals, ordered A090181.

%Y Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.

%K nonn

%O 1,4

%A _Gus Wiseman_, Nov 29 2022

%E Terms a(19) and beyond from _Andrew Howroyd_, Jan 01 2023