We propose a homotopy algebra associated to classical open-closed strings. We call it an {\em open-closed homotopy algebra.} It is an extension of an $L_\infty$-algebra by an $A_\infty$-algebra. Thus, it includes an $L_\infty$-algebra and an $A_\infty$-algebra as substructures. In general, tree (=classical) open strings and closed strings have an $A_\infty$-algebraic structure and an $L_\infty$-structures, respectively. We explain that our open-closed homotopy algebraic structure is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory in an appropriate way. Namely, this is a homotopy algebraic structure which a classical open-closed string field theory has. We explain this open-closed homotopy algebraic structure appears in various other mathematical physics of tree open-closed strings such as Kontsevich's deformation quantization. In such contexts, our open-closed homotopy algebra gives us a general scheme for deformation of open string structures ($A_\infty$-algebras) by closed strings ($L_\infty$-algebras). We show that this open-closed homotopy algebra is actually a homotopy invariant notion; for instance the minimal model theorem holds.