It Depends On What Else She's Doing and if She Has enough Movement Remaining
If she has taken an Action and then leaves someone's reach without using the Disengage action, she'll provoke an opportunity attack (per Dale M's answerper Dale M's answer).
PHB p. 192. Disengage
If you take the Disengage action, your movement doesn't provoke opportunity attacks for the rest of your turn.
Reviewing the Dryad description on MM Page p. 121, her Actions are listed as a Club (melee weapon attack) and a Fey Charm. That tree stride looks to be a subset of her movement.(Not an action, not a bonus action).
Therefore, if she takes the Disengage action (rather than an attack or an attempt at Fey Charm) and then uses 10' of movement (providing she has that much movement left for that turn) she'd be able to avoid the Opportunity Attack and go through the trees.