What was the relationship between the houses of Plantagenet, Lancaster, and York during the War of the Roses?
The House of Plantagenet and its two branches described below ruled England for over three hundred years (1154-1485.) Both the House of Lancaster and House of York were descended from the Plantagenets which gave both a nominal claim to the throne. The Lancastrians claimed the throne through Edward III's third son, John of Gault; the Yorkists through Edward III's second son, although the claim was through his mother's line. Although there had been intermittent skirmishes, the issue of right to the throne erupted in earnest after England's loss of the Hundred Years War to France. The House of York was represented by a white rose on its coat of arms; the House of Lancaster by a red rose, so the intermittent battles between the two became known as the Wars of the Roses. England was ruled by three Lancastrian Kings, Henry IV, Henry V, and Henry VI; followed by three Yorkist Kings, Edward IV, Edward V, and Richard III. After a period of some military upheaval, Edward IV was crowned in 1451. Edward died suddenly in 1453 and his son, Edward V was proclaimed King. He was a thirteen at the time, and his uncle, Richard of Gloucester and brother of Edward IV ruled as his regent. Ultimately, Richard deposed the young king who was murdered in the Tower of London. The perpetrators of the murder are to still a matter of intense debate. Richard had himself crowned Richard III.
Henry Tudor, a descendant of the Lancastrian line, claimed that Richard had usurped the throne, and that he (Henry) was the rightful claimant. TheWars were finally settled at the Battle of Bosworth Field when Richard was killed and Henry Tudor was crowned Henry VII.
Two excellent resources you might consider are both by Alison Weir, a superb Tudor Historian: The Princes in the Tower; and The Wars of the Roses.