• DOI:10.1017/s0963548322000128
• Corpus ID: 248069676

On the peel number and the leaf-height of Galton–Watson trees

@article{Devroye2021OnTP,
  title={On the peel number and the leaf-height of Galton–Watson trees},
  author={Luc Devroye and Marcel K. Goh and Rosie Y. Zhao},
  journal={Combinatorics, Probability and Computing},
  year={2021}
}

• L. Devroye, Marcel K. Goh, Rosie Y. Zhao
• Published 28 June 2021
• Mathematics
• Combinatorics, Probability and Computing

<jats:p>We study several parameters of a random Bienaymé–Galton–Watson tree <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline1.png" />
<jats:tex-math>
$T_n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> of size <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline2.png" />
<jats:tex-math>
$n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> defined in terms of an offspring distribution <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline3.png" />
<jats:tex-math>
$\xi$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with mean <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline4.png" />
<jats:tex-math>
$1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and nonzero finite variance <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline5.png" />
<jats:tex-math>
$\sigma ^2$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. Let <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline6.png" />
<jats:tex-math>
$f(s)=\mathbb{E}\{s^\xi \}$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> be the generating function of the random variable <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline7.png" />
<jats:tex-math>
$\xi$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. We show that the independence number is in probability asymptotic to <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline8.png" />
<jats:tex-math>
$qn$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, where <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline9.png" />
<jats:tex-math>
$q$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is the unique solution to <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline10.png" />
<jats:tex-math>
$q = f(1-q)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. One of the many algorithms for finding the largest independent set of nodes uses a notion of repeated peeling away of all leaves and their parents. The number of rounds of peeling is shown to be in probability asymptotic to <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline11.png" />
<jats:tex-math>
$\log n/\log (1/f'(1-q))$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. Finally, we study a related parameter which we call the leaf-height. Also sometimes called the protection number, this is the maximal shortest path length between any node and a leaf in its subtree. If <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline12.png" />
<jats:tex-math>
$p_1 = \mathbb{P}\{\xi =1\}\gt 0$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, then we show that the maximum leaf-height over all nodes in <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline13.png" />
<jats:tex-math>
$T_n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is in probability asymptotic to <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline14.png" />
<jats:tex-math>
$\log n/\log (1/p_1)$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. If <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline15.png" />
<jats:tex-math>
$p_1 = 0$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline16.png" />
<jats:tex-math>
$\kappa$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is the first integer <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline17.png" />
<jats:tex-math>
$i\gt 1$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline18.png" />
<jats:tex-math>
$\mathbb{P}\{\xi =i\}\gt 0$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, then the leaf-height is in probability asymptotic to <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0963548322000128_inline19.png" />
<jats:tex-math>
$\log _\kappa \log n$
</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>.</jats:p>