Bounding mean orders of sub-kk-trees of kk-trees

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.Comment: 20 Pages, 6 Figure

On the distribution of subtree orders of a tree

CITATION: Ralaivaosaona, D. & Wagner, S. 2018. On the distribution of subtree orders of a tree. Ars Mathematica Contemporanea, 14(1):129-156, doi:10.26493/1855-3974.996.675.The original publication is available at https://amc-journal.euWe investigate the distribution of the number of vertices of a randomly chosen subtree of a tree. Specifically, it is proven that this distribution is close to a Gaussian distribution in an explicitly quantifiable way if the tree has sufficiently many leaves and no long branchless paths. We also show that the conditions are satisfied asymptotically almost surely for random trees. If the conditions are violated, however, we exhibit by means of explicit counterexamples that many other (non-Gaussian) distributions can occur in the limit. These examples also show that our conditions are essentially best possible.https://amc-journal.eu/index.php/amc/article/view/996Publisher's versio