Summary: Given a weighted graph \(G\) on \(n+1\) vertices, a spanning \(K\)-tree \(T_K\) of \(G\) is defined to be a spanning tree \(T\) of \(G\) together with \(K\) distinct edges of \(G\) that are not edges of \(T\). The objective of the minimum-cost spanning \(K\)-tree problem is to choose a subset of edges to form a spanning \(K\)-tree with the minimum weight. In this paper, we consider the constructing spanning K-tree problem that is a generalization of the minimum-cost spanning \(K\)-tree problem. We are required to construct a spanning \(K\)-tree \(T_K\) whose \(n+K\) edges are assembled from some stock pieces of bounded length \(L\). Let \(c_0\) be the sale price of each stock piece of length \(L\) and \(k(T_K)\) the number of minimum stock pieces to construct the \(n+K\) edges in \(T_K\). For each edge \(e\) in \(G\), let \(c(e)\) be the construction cost of that edge \(e\). Our new objective is to minimize the total cost of constructing a spanning \(K\)-tree \(T_K\), i.e., \(\min_{T_K}\{\sum_{e\in T_K}c(e)+k(T_K)\cdot c_0\}\). The main results obtained in this paper are as follows. (1) A 2-approximation algorithm to solve the constructing spanning \(K\)-tree problem. (2) A \(\frac{3}{2}\)-approximation algorithm to solve the special case for constant construction cost of edges. (3) An APTAS for this special case.