The Colorado Avalanche have the best chance of winning picks one through three at the 2017 NHL Draft. The farthest they could fall would be pick #4.
Historically the team doesn’t like to pick defensemen too high. The last time they chose a defenseman in the first round was 2011, when they selected Duncan Siemens 11th overall. That probably didn’t sweeten them up to the idea considering he’s only played four games in the NHL since then.
Nonetheless, the 2017 NHL draft class has a highly-rated defenseman prospect, Swede Timothy Liljegren. The Colorado Avalanche are desperate for defense, so you never know if they might try another gamble in the first round.