2005 International Conference on Analysis of Algorithms
Conrado Martínez (ed.)
DMTCS Conference Volume AD (2005), pp. 223230
author:  Gerard Kok 

title:  Pattern distribution in various types of random trees 
keywords:  random trees, generating functions, limiting distributions 
abstract: 
Let
T
denote the set of unrooted unlabeled trees of size
n
n
and let
M
be a particular (finite) tree. Assuming that every
tree of
k
T
is equally likely, it is shown that the number of
occurrences
n
X
of
n
M
as an induced subtree satisfies
k
E X
and
n
∼µn
Var X
for some (computable) constants
n
∼σ
2
n
µ> 0
and
σ≥0
. Furthermore, if
σ>0
then
(X
converges to a limiting distribution with density
n
 E X
n
)/√
Var
X
n
(A+Bt
for some constants
2
)e
Ct
2
A,B,C
. However, in all cases in which we were able to
calculate these constants, we obtained
B=0
and thus a normal distribution. Further, if we
consider planted or rooted trees instead of
T
then the limiting distribution is always normal.
Similar results can be proved for planar, labeled and
simply generated trees.
n

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reference:  Gerard Kok (2005), Pattern distribution in various types of random trees, in 2005 International Conference on Analysis of Algorithms, Conrado Martínez (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AD, pp. 223230 
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