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. A rooted tree determines an oriented tree by directing every edge away from the root vertex (Knuth 1997, pp. 385-399). An unrooted tree has no distinguished root vertex and is also called a free tree. The unqualified term "tree" generally refers to an unrooted tree.
A graph disjoint union of rooted trees is a rooted forest.
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, counted up to root-preserving
graph isomorphism, 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 
. The generating function
is
 |  |  |
(1)
|
 |  |  |
(2)
|
This power series satisfies
 |  | ![xexp[sum_(r=1)^(infty)1/rT(x^r)]](/images/equations/RootedTree/Inline13.svg) |
(3)
|
 |  | ![T(x)-1/2[T^2(x)-T(x^2)],](/images/equations/RootedTree/Inline16.svg) |
(4)
|
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)
|
for 
, with 
and 
, where the second sum is over the positive
divisors 
of 
(Finch 2003).
As shown by Otter (1948),
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A051491; Odlyzko 1995; Knuth 1997, p. 396), where 
is given by the unique positive root
of
 |
(9)
|
An asymptotic series for 
is given by
 |
(10)
|
where the constants can be computed in terms of partial derivatives of the function
![F(x,y)=xexp[y+sum_(k=2)^infty(T(x^k))/k]-y](/images/equations/RootedTree/NumberedEquation5.svg) |
(11)
|
(Plotkin and Rosenthal 1994, Finch 2019).
