The paper presents a direct proof of the tree-level recursion relation for scattering amplitudes in Yang-Mills theory, based solely on the properties of tree-level amplitudes. The authors, Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten, introduce a new set of recursion relations that express any tree-level amplitude of gluons as a sum over terms constructed from the product of two subamplitudes with fewer gluons times a Feynman propagator. These subamplitudes are physical, on-shell amplitudes with shifted momenta. The recursion relations are derived using the known properties of one-loop amplitudes and are shown to be valid for reference gluons of both opposite and the same helicity. The proof relies on basic facts about tree diagrams, such as the singularities coming from poles of internal propagators, and the description of tree amplitudes via MHV diagrams. The authors demonstrate that the recursion relations can be defined for reference gluons of the same helicity and that the gluons do not need to be adjacent. They also show that MHV tree diagrams give the same Yang-Mills tree amplitudes as Feynman diagrams. The paper includes a detailed derivation of the BCF recursion relations using the spinor-helicity formalism and a proof that the generalized amplitude \( A(z) \) vanishes at infinity for specific helicity configurations. This proof is achieved using both MHV tree diagrams and standard Feynman diagrams. The authors conclude by discussing the implications of their results, emphasizing the convenience of the BCF recursion relations for determining Yang-Mills tree amplitudes from their singularities.The paper presents a direct proof of the tree-level recursion relation for scattering amplitudes in Yang-Mills theory, based solely on the properties of tree-level amplitudes. The authors, Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten, introduce a new set of recursion relations that express any tree-level amplitude of gluons as a sum over terms constructed from the product of two subamplitudes with fewer gluons times a Feynman propagator. These subamplitudes are physical, on-shell amplitudes with shifted momenta. The recursion relations are derived using the known properties of one-loop amplitudes and are shown to be valid for reference gluons of both opposite and the same helicity. The proof relies on basic facts about tree diagrams, such as the singularities coming from poles of internal propagators, and the description of tree amplitudes via MHV diagrams. The authors demonstrate that the recursion relations can be defined for reference gluons of the same helicity and that the gluons do not need to be adjacent. They also show that MHV tree diagrams give the same Yang-Mills tree amplitudes as Feynman diagrams. The paper includes a detailed derivation of the BCF recursion relations using the spinor-helicity formalism and a proof that the generalized amplitude \( A(z) \) vanishes at infinity for specific helicity configurations. This proof is achieved using both MHV tree diagrams and standard Feynman diagrams. The authors conclude by discussing the implications of their results, emphasizing the convenience of the BCF recursion relations for determining Yang-Mills tree amplitudes from their singularities.