| 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 sub-tree 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. 223-230 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| pdf-source: | dmAD0121.pdf (123 K) |
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