The House of Beaufort’s claim to the throne is based on their direct male line descent from John of Gaunt, the 3rd son of Edward III; however the taint of illegitimacy haunted their cause. The Beaufort’s were descended from John Beaufort, son of John of Gaunt and his third wife Katherine Swynford, but John and his siblings had all been born before the couple were married. Richard II and an Act of Parliament formally legitimized the Beaufort children, however their half-brother Henry IV issued a proclamation that barred the Beauforts from the succession but without an Act of Parliament making its lawfulness questionable. Nevertheless, Edmund Beaufort saw it has his right to succeed Henry VI since like Henry IV before because he was the senior direct male descendant of Edward III and no one could argue the point.
The House of York’s claim to the throne was also based on descent from Edward III as well, but twice over. The Dukes of York were direct male descendants of Edward III through his 4th son, Edmund of Langley and first Duke of York, and unlike the Beaufort claim to direct lineage, the House of York did not have the taint of illegitimacy hanging over their family. But to Richard Plantagenet what gave him the right to the throne was that he was a descended from Edward III’s 2nd son, Lionel of Antwrep. Lionel only had one surviving child, Philippa of Clarence, the 5th Countess of Ulster, who married Edmund Mortimer, the 3rd Earl of March. During her lifetime, Philippa was the heir presumptive to her cousin Richard II and upon her death that right passed to her son Roger Mortimer, the 4th Earl of March. When Roger Mortimer died in July 1398, he left four children including two sons the oldest of which was his namesake the six-year old Roger who inherited his grandmother’s position of heir presumptive. When Richard II “resigned” the throne in 1399, the seven-year old Roger Mortimer’s right was pushed aside in favor of Henry of Bolingbroke who became Henry IV and began the House of Lancaster. The 5th Earl of March lived until 1425 but left no surviving issue and since his younger brother had died young, the Mortimer rights to the throne were passed to Richard Plantagenet, the son of his older sister Anne Mortimer. As the House of Plantagenet inherited the throne through Matilda, the daughter of Henry I of England, who’s right to rule had been usurped by King Stephen but was reestablished by her son the first Plantagenet King of England, Henry II.
Because of the events of 1399 and Richard II’s “resignation,” both the House of Beaufort and the House of York seemed to have clear claims to the Crown. The Beauforts held the agnatic claim while the Yorkists were the legitimist claim, while today the claim of the House of York would be paramount in law back in 1453 it was still an open question. And like many other legal arguments of so high importance back then, the issue would need to be decided not in Parliament or the court room but the battlefield.