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Asymptotical Growth of a Class of Random Trees

B. Pittel
The Annals of Probability
Vol. 13, No. 2 (May, 1985), pp. 414-427
Stable URL: http://www.jstor.org/stable/2243800
Page Count: 14
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Asymptotical Growth of a Class of Random Trees
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Abstract

We study three rules for the development of a sequence of finite subtrees {tn} of an infinite m-ary tree t. Independent realizations {ω(n)} of a stationary ergodic process {ω} on m letters are used to trace out paths in t. In the first rule, tn is formed by adding a node to tn - 1 at the first location where the path defined by ω (n) leaves tn - 1. The second and third rules are similar, but more complicated. For each rule, the height Ln 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 σ1, σ2, 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 tn grows as σ2ln n with probability one.

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