DMTCS Proceedings, 2005 International Conference on Analysis of Algorithms

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DMTCS Conference vol AD (2005), pp. 223-230

DMTCS

2005 International Conference on Analysis of Algorithms

Conrado Martínez (ed.)

DMTCS Conference Volume AD (2005), pp. 223-230


author: Gerard Kok
title: Pattern distribution in various types of random trees
keywords: random trees, generating functions, limiting distributions
abstract: Let
T
n
denote the set of unrooted unlabeled trees of size
n
and let
M
k
be a particular (finite) tree. Assuming that every tree of
T
n
is equally likely, it is shown that the number of occurrences
X
n
of
M
k
as an induced sub-tree satisfies
E X
n
∼µn
and
Var X
n
∼σ
2
n
for some (computable) constants
µ> 0
and
σ≥0
. Furthermore, if
σ>0
then
(X
n
- E X
n
)/√
Var
X
n
converges to a limiting distribution with density
(A+Bt
2
)e
-Ct
2
for some constants
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
n
then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.
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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. 223-230
bibtex: For a corresponding BibTeX entry, please consider our BibTeX-file.
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