In order to understand the NHL playoffs (aka post season), it is important to understand the regular season and the structure of the league as both of these factors greatly affect the teams included in the playoffs as well as the playoff tree or "seeding." The playoff tree is the schedule of which teams play each other during the post season. Seeding is the placement of teams based on ranking.
The hockey teams in the NHL are divided into two conferences: the Eastern Conference and the Western Conference. Both conferences are broken down further into divisions based on geographic location. The East's divisions are Atlantic, Northeast, and Southeast. Similarly, the West's divisions are Central, Northwest, and Pacific. Each division has five teams totaling 30 teams in the entire league.
Which teams make the playoffs are determined by a system based on points. During the 82 game regular season, teams score points based on individual games. If a team wins a game, they are awarded two points. If a team loses a game, they are not awarded any points. If a team loses a game that went into over time, they are awarded one point. The eight teams with the most points in each conference at the end of the regular season makes the playoffs.
It is possible for a top eight team to not make the playoffs, but this event is very unusual. Every division has a 1st place team meaning they have the most points in their division. Every division winner makes the playoffs regardless of their overall rank in their conference. For example, the Edmonton Oilers could win the Northwest division with a 9th place ranking in the West, and they would still make the playoffs.
Once the teams that make the playoffs have been decided, the seeding determines which teams play one another in the post season. Within a conference the 1st place team plays the 8th place team, the 2nd place team plays the 7th place team, etc. This series of games is called the 1st round of the playoffs.
Each round of the playoffs is a best out of seven series. Another way to think of it is the first team to win four games in a round of playoffs moves on to the next round; the losing team is eliminated. Once a round of playoffs is completed, the teams are re-seeded. The highest ranking team plays the lowest ranking team left. The second highest ranking team plays the second lowest ranking team, etc.
An important aspect of seeding is the advantage of playing at home. Games one, two, five, and seven of a series are played in the city of the higher ranking team. Games three, four, and six are played in the city of the lower ranking team. Playing on the road can be difficult for a visiting team because of the fans, jet lag, anxiety, arena familiarity, and other factors.
Eventually, all the teams are eliminated except for two teams: the Eastern Conference champions and the Western Conference champions. They play in a series referred to as the Stanley Cup Finals. Home ice is determined again by seeding. The winner of the Stanley Cup Finals is awarded the Stanley Cup and are considered the champions of the NHL. The names of the players on the winning team are engraved on the Stanley Cup which passes to the new winners every season.

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