## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 2 (1985), 319-637

### Asymptotical Growth of a Class of Random Trees

#### Abstract

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

**Source**

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

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176993000

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176993000

**Mathematical Reviews number (MathSciNet)**

MR781414

**Zentralblatt MATH identifier**

0563.60010

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60F15: Strong theorems 28D20: Entropy and other invariants 68C25

**Keywords**

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

#### Citation

Pittel, B. Asymptotical Growth of a Class of Random Trees. The Annals of Probability 13 (1985), no. 2, 414--427. doi:10.1214/aop/1176993000. http://projecteuclid.org/euclid.aop/1176993000.