Bounds on Sweep-Covers by Raney Numbers

In this work, we introduce a vertex separator in trees known as a sweep-cover that is defined by an ancestor-descendent relationship with all nodes in the tree. We prove the recurrence relation of sweep-covers with $n$ subcovers $P_{\Delta, \gamma}(n)$ on a class of infinite $\Delta$-ary trees with constant path lengths $\gamma$ between the $\Delta$-star internal nodes. Then, we provide recurrence relations for Raney numbers over integer compositions and show that they provide a lower-bound for sweep-covers such that $P_{\Delta, \gamma}(n) = \Omega\left( \frac{\sqrt{2 \pi} n^{\Delta n + \Delta + \frac{3}{2}}}{e^n ((\Delta-1)n+\Delta+1)!(n+1)!} \gamma \right)$.