In　this　paper,　we　study　the　prize-collecting　k-Steiner　tree　problem　(PC　k-ST),　which　is　an　interesting　generalization　of　both　the　k-Steiner　tree　problem　(k-ST)　and　the　prize-collecting　Steiner　tree　problem　(PCST).　In　the　PC　k-ST,　we　are　given　an　undirected　connected　graph　G=　(V,　E),　a　subset　R⊆　V　called　terminals,　a　root　vertex　r∈　V　and　an　integer　k.　Every　edge　has　a　non-negative　edge　cost　and　every　vertex　has　a　non-negative　penalty　cost.　We　wish　to　find　an　r-rooted　tree　F　that　spans　at　least　k　vertices　in　R　so　as　to　minimize　the　total　edge　costs　of　F　as　well　as　the　penalty　costs　of　the　vertices　not　in　F.　As　our　main　contribution,　we　propose　two　approximation　algorithms　for　the　PC　k-ST　with　ratios　of　5.9672　and　5.　The　first　algorithm　is　based　on　an　observation　of　the　solutions　for　the　k-ST　and　the　PCST,　and　the　second　one　is　based　on　the　technique　of　primal-dual.　©　2021,　Springer　Nature　Switzerland　AG.