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{\bf Markus Kuba and Alois Panholzer}
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{\bf Descendants in Increasing Trees}
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Simple families of increasing trees can be constructed from simply
generated tree families, if one considers for every tree of size $n$
all its increasing labellings, i.$\,$e. labellings of the nodes by
distinct integers of the set $\{1, \dots, n\}$ in such a way that each
sequence of labels along any branch starting at the root is
increasing. Three such tree families are of particular interest:
{\it recursive trees}, {\it plane-oriented recursive trees} and
{\it binary increasing trees}. We study the quantity {\it number of
descendants of node $j$ in a random tree of size $n$} and give closed
formul{\ae} for the probability distribution and all factorial moments
for those subclass of tree families, which can be constructed via an
insertion process. Furthermore limiting distribution results of this
parameter are given.
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