A rooted tree is a tree in which a special ("labeled") node is singled out. This node is called the "root "
or (less commonly) "eve" of the tree. Rooted trees are equivalent to oriented
trees (Knuth 1997, pp. 385-399). A tree which is not rooted is sometimes called
a free tree , although the unqualified term "tree"
generally refers to a free tree.
A rooted tree in which the root vertex has vertex
degree 1 is known as a planted tree .
The numbers of rooted trees on A000081 ). Denote the number
of rooted trees with generating function is
This power series satisfies
where generating function for unrooted trees . A generating function
for the sequence itself as
(5)
The number of rooted trees can also be calculated from the recurrence
relation
(6)
with divide
As shown by Otter (1948),
(OEIS A051491 ; Odlyzko 1995; Knuth 1997, p. 396), where positive root
of
(9)
If asymptotic series for
(10)
where the constants can be computed in terms of partial derivatives of the function
(11)
(Plotkin and Rosenthal 1994; Finch 2003).
See also Free Tree ,
Ordered Tree ,
Planted Tree ,
Red-Black
Tree ,
Rooted Graph ,
Tree ,
Weakly Binary Tree
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References Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Finch, S. R. "Otter's Tree Enumeration
Constants." §5.6 in Mathematical
Constants. Finch, S. "Two Asymptotic Series." December 10, 2003.
http://algo.inria.fr/bsolve/ . Harary,
F. Graph
Theory. Harary,
F. and Palmer, E. M. "Rooted Trees." §3.1 in Graphical
Enumeration. Knuth,
D. E. The
Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Nijenhuis, A. and Wilf, H. Combinatorial
Algorithms for Computers and Calculators, 2nd ed. Odlyzko, A. M. "Asymptotic Enumeration Methods."
In Handbook
of Combinatorics, Vol. 2 http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf . Otter,
R. "The Number of Trees." Ann. Math. 49 , 583-599, 1948. Plotkin,
J. M. and Rosenthal, J. W. "How to Obtain an Asymptotic Expansion
of a Sequence from an Analytic Identity Satisfied by Its Generating Function."
J. Austral. Math. Soc. Ser. A 56 , 131-143, 1994. Pólya,
G. "On Picture-Writing." Amer. Math. Monthly 63 , 689-697,
1956. Ruskey, F. "Information on Rooted Trees." http://www.theory.csc.uvic.ca/~cos/inf/tree/RootedTree.html . Sloane,
N. J. A. Sequences A000081 /M1180
and A051491 in "The On-Line Encyclopedia
of Integer Sequences." Wilf, H. S. Combinatorial
Algorithms: An Update. Referenced
on Wolfram|Alpha Rooted Tree
Cite this as:
Weisstein, Eric W. "Rooted Tree." From
MathWorld https://mathworld.wolfram.com/RootedTree.html
Subject classifications
nodes for 
, 2, ... are 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842,
4766, ... (OEIS A000081). Denote the number
of rooted trees with 
nodes by 
,
then the generating function is
![xexp[sum_(r=1)^(infty)1/rT(x^r)]](/images/equations/RootedTree/Inline13.svg)
![T(x)-1/2[T^2(x)-T(x^2)],](/images/equations/RootedTree/Inline16.svg)
is the generating function for unrooted trees. A generating function
for 
can be written using a product involving
the sequence itself as
and 
, where the second sum is over all 
which divide 
(Finch 2003).
is given by the unique positive root
of
is the number of nonisomorphic rooted
trees on 
nodes, then an asymptotic series for 
is given by
![F(x,y)=xexp[y+sum_(k=2)^infty(T(x^k))/k]-y](/images/equations/RootedTree/NumberedEquation5.svg)
