The mean order of sub‐<italic>k</italic>‐trees of <italic>k</italic>‐trees.

Abstract: This article focuses on the problem of determining the mean orders of sub‐<italic>k</italic>‐trees of <italic>k</italic>‐trees. It is shown that the problem of finding the mean order of all sub‐<italic>k</italic>‐trees containing a given <italic>k</italic>‐clique <italic>C</italic>, 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‐<italic>k</italic>‐trees containing a given sub‐<italic>k</italic>‐tree. The second extension focuses on the expected number of <italic>r</italic>‐cliques, 1 ≤ r ≤ k + 1, in a randomly chosen sub‐<italic>k</italic>‐tree containing a fixed sub‐<italic>k</italic>‐tree <italic>X</italic>. Sharp lower bounds for both invariants are derived. The article concludes with a study of global mean orders of sub‐<italic>k</italic>‐trees of a <italic>k</italic>‐tree. For a <italic>k</italic>‐tree, from the class of simple‐clique <italic>k</italic>‐trees, it is shown that the mean order of its sub‐<italic>k</italic>‐trees is asymptotically equal to the mean subtree order of its dual. For general <italic>k</italic>‐trees a recursive generating function for the number of sub‐<italic>k</italic>‐trees of a given <italic>k</italic>‐tree <italic>T</italic> is derived. [ABSTRACT FROM AUTHOR]