Rooted Tree

Author

Eric W. Weisstein
August 15, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/RootedTree.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RootedTree.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Counts

Length/@RootedTrees/@Range[12]//Timing

Generation

RootedTrees[n=10]//Length//Timing

RootedTrees[n=11]//Length//Timing

RootedTrees[n=12]//Length//Timing

Recurrence

T[0]:=0
T[1]:=1
T[n_]:=T[n]=Plus@@Table[T[n-j]Plus@@(# T[#]&/@Divisors[j]),{j,n-1}]/(n-1)

T/@Range[20]

Constant

Polynomial[l_List, x_] := Fold[x #1 + #2 &, 0, l]

T[x_,n_:30]:=Expand[x Polynomial[T/@Range[n,1,-1],x]]

Block[{$RecursionLimit=1000},
x/.FindRoot[T[1/x,300]==1,{x,2.9,3},WorkingPrecision->20]
]//Timing

Block[{$RecursionLimit=2000},
With[{n=500},
N[T[n]/T[n-1],20]
]
]//Timing

Actual value is 2.95576528565199497…

Enumeration

ShowGraphArray[Flatten[Map[Partition[#,9,9,{1,1},{}]&,RootedTrees/@Range[5],{1}],1]
]