The Annals of Probability

Asymptotical Growth of a Class of Random Trees

B. Pittel

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We study three rules for the development of a sequence of finite subtrees $\{t_n\}$ of an infinite $m$-ary tree $t$. Independent realizations $\{\omega(n)\}$ of a stationary ergodic process $\{\omega\}$ on $m$ letters are used to trace out paths in $t$. In the first rule, $t_n$ is formed by adding a node to $t_{n - 1}$ at the first location where the path defined by $\omega (n)$ leaves $t_{n - 1}$. The second and third rules are similar, but more complicated. For each rule, the height $L_n$ of the added node is shown to grow, in probability, as $\ln n$ divided by $h$ the entropy per symbol of the generic process. A typical retrieval time has the same behavior. On the other hand, $\lim \inf_nL_n/\ln n = \sigma_1, \lim \sup_n L_n/\ln n = \sigma_2$ a.s., where the constants $\sigma_1, \sigma_2$, are, in general, different, depend on the rule in use, and $\sigma_1 < 1/h < \sigma_2$. It is proven along the way that the height of $t_n$ grows as $\sigma_2\ln n$ with probability one.

Article information

Ann. Probab. Volume 13, Number 2 (1985), 414-427.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60C05: Combinatorial probability
Secondary: 60F15: Strong theorems 28D20: Entropy and other invariants 68C25

Random trees lengths of the paths ergodic process asymptotic growth strong weak convergence


Pittel, B. Asymptotical Growth of a Class of Random Trees. Ann. Probab. 13 (1985), no. 2, 414--427. doi:10.1214/aop/1176993000.

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