A vertex of degree one is called an end-vertex and the set of end-vertices of G is denoted by End(G). For a positive integer k, a tree T be called k-ended tree if $|End(T)| \leq k$. In this paper, we obtain sufficient conditions for spanning k-trees of 3-regular connected graphs. We give a construction sequence of graphs satisfying the condition. At the end, we present a conjecture about spanning k-ended trees of 3-regular connected graphs.

The Stirling number of the second kind \( S(n, k) \) counts the number of ways to partition a set of \( n \) labeled balls into \( k \) non-empty unlabeled cells. We extend this problem and give a new statement of the \( r \)-Stirling numbers of the second kind and \( r \)-Bell numbers. We also introduce the \( r \)-mixed Stirling number of the second kind and \( r \)-mixed Bell numbers. As an application of our results we obtain a formula for the number of ways to write an integer \( m > 0 \) in the form \( m_1 \cdot m_2 \cdot \cdots \cdot m_k \), where \( k \geq 1 \) and \( m_i \)'s are positive integers greater than 1.