2 research outputs found
Fringe trees for random trees with given vertex degrees
We prove asymptotic normality for the number of fringe subtrees isomorphic to
any given tree in uniformly random trees with given vertex degrees. As
applications, we also prove corresponding results for random labelled trees
with given vertex degrees, for random simply generated trees (or conditioned
Galton--Watson trees), and for additive functionals.
The key tool for our work is an extension to the multivariate setting of a
theorem by Gao and Wormald (2004), which provides a way to show asymptotic
normality by analysing the behaviour of sufficiently high factorial moments.Comment: 41 page
Central limit theorems for additive functionals and fringe trees in tries
We give general theorems on asymptotic normality for additive functionals of random tries generated by a sequence of independent strings. These theorems are applied to show asymptotic normality of the distribution of random fringe trees in a random trie. Formulas for asymptotic mean and variance are given. In particular, the proportion of fringe trees of size k (defined as number of keys) is asymptotically, ignoring oscillations, c/(k(k - 1)) for k >= 2, where c = 1/(1 + H) with H the entropy of the letters. Another application gives asymptotic normality of the number of k-protected nodes in a random trie. For symmetric tries, it is shown that the asymptotic proportion of k-protected nodes (ignoring oscillations) decreases geometrically as k -> infinity