Longest paths in random apollonian networks and largest r-ary subtrees of random d-ary recursive trees

Let r and d be positive integers with r < d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let T_d,t be the tree produced after t steps. We show that there exists a fixed δ < 1 depending on d and r such that almost surely for all large t , every r-ary subtree of T_d,t has less than t^δ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t + 3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ < 1, such that eventually every path in this graph has length less than t^δ , which verifies a conjecture of Cooper and Frieze (2015).