The mean order of sub-k-trees of k-trees

This article focuses on the problem of determining the mean orders of sub-k-trees of k-trees. It is shown that the problem of finding the mean order of all sub-k-trees containing a given k-clique C, can be reduced to the previously studied problem of finding the mean order of subtrees of a tree that contain a given vertex. This problem is extended in two ways. The first of these extensions focuses on the mean order of sub-k-trees containing a given sub-k-tree. The second extension focuses on the expected number of r-cliques, 1rk+1, in a randomly chosen sub-k-tree containing a fixed sub-k-tree X. Sharp lower bounds for both invariants are derived. The article concludes with a study of global mean orders of sub-k-trees of a k-tree. For a k-tree, from the class of simple-clique k-trees, it is shown that the mean order of its sub-k-trees is asymptotically equal to the mean subtree order of its dual. For general k-trees a recursive generating function for the number of sub-k-trees of a given k-tree T is derived.