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 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

This power series satisfies

where
is the generating function for unrooted trees . A generating function
for can be written using a product involving
the sequence itself as

(5)

The number of rooted trees can also be calculated from the recurrence
relation

(6)

with and , where the second sum is over all which divide (Finch 2003).

As shown by Otter (1948),

(OEIS A051491 ; Odlyzko 1995; Knuth 1997, p. 396), where
is given by the unique positive root
of

(9)

If is the number of nonisomorphic rooted
trees on
nodes, then an asymptotic series for is given by

(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. Wellesley, MA: A
K Peters, p. 22, 2003. Finch, S. R. "Otter's Tree Enumeration
Constants." §5.6 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 295-316,
2003. Finch, S. "Two Asymptotic Series." December 10, 2003.
http://algo.inria.fr/bsolve/ . Harary,
F. Graph
Theory. Reading, MA: Addison-Wesley, pp. 187-190 and 232, 1994. Harary,
F. and Palmer, E. M. "Rooted Trees." §3.1 in Graphical
Enumeration. New York: Academic Press, pp. 51-54, 1973. Knuth,
D. E. The
Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed.
Reading, MA: Addison-Wesley, 1997. Nijenhuis, A. and Wilf, H. Combinatorial
Algorithms for Computers and Calculators, 2nd ed. New York: Academic Press,
1978. Odlyzko, A. M. "Asymptotic Enumeration Methods."
In Handbook
of Combinatorics, Vol. 2 (Ed. R. L. Graham, M. Grötschel,
and L. Lovász). Cambridge, MA: MIT Press, pp. 1063-1229, 1995. 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. Philadelphia, PA: SIAM, 1989. Referenced
on Wolfram|Alpha Rooted Tree
Cite this as:
Weisstein, Eric W. "Rooted Tree." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RootedTree.html

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