Tanev, who had received interest from teams overseas, recorded two goals and five assists for seven points (2-5-7) and 10 penalty minutes in 38 games with Vancouver last season.
In 92 NHL contests with Vancouver, the 23-year-old has collected two goals and eight assists for 10 points and 12 penalty minutes.
The Toronto native was originally signed by Vancouver as a free agent May 31, 2010.
Alberts, 32, appeared in 24 games last season, recording one assist, 32 penalty minutes and 39 hits.
He has appeared in 449 career games and has accumulated eight goals and 47 assists for 55 points and 492 penalty minutes over the span of eight NHL seasons with the Boston Bruins, Philadelphia Flyers, Carolina Hurricanes and Canucks.
The Minneapolis native was originally selected 179th overall in the 2001 NHL Entry Draft by the Boston Bruins.