Increasing paths in regular trees
نویسندگان
چکیده
We consider a regular n-ary tree of height h, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of simple paths from the root to a leaf along vertices with increasing labels. We show that if α = n/h is fixed and α > 1/e, the probability there exists such a path converges to 1 as h → ∞. This complements a previously known result that the probability converges to 0 if α 6 1/e.
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