YES the Dryad does provoke an Opportunity Attack, given she has not used the Disengage action or any other ability that avoids triggering opportunity attacks.
You could argue that the Dryad teleports as part of entering the tree, that she is not there to hit and that teleportation does not provoke, or perhaps the tree is in the way. This would be an incorrect application of the rules to the order of events:
- the Dryad moves, using up 10' of her movement, into the space of a tree within her reach
- if this move would put her out of your reach this provokes an opportunity attack which interrupts her movement, it happens before she leaves your reach, i.e. before she teleports, before the tree could be in the way
- if you have your reaction available you can make an opportunity attack
- if she is still able to, after the results of the opportunity attack, the Dryad completes her move into the space of the tree
- the Dryad teleports
The important factors are: that the Dryad has to use some of her move (magical or not) to enter the tree, which if takes her out of your reach satisfies the requirements to trigger an opportunity attack; and that the opportunity attack interrupts and happens before she leaves your reach, i.e. before she enters the tree.
As the teleport part of this ability has her appear within 5' of the tree, she does not move out of the tree at the other end and so this does not provoke an opportunity attack. However any of her subsequent movement (remaining after subtracting the 10') is as normal.
As this ability is listed as using part of her movement she can use the disengage action first (obviously first because it covers movement during the rest of your turn) to avoid this consequence of her move into the tree, and any movement after she has re-appeared.
Opportunity Attack (PHB p. 195):
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
The attack interrupts the provoking creature’s movement, occurring right before the creature leaves your reach.
You also don’t provoke an opportunity attack when you teleport
Tree Stride (MM p.121):
Once on her turn, the dryad can use 10 feet of her movement to step magically into one living tree within her reach and emerge from a second living tree within 60 feet of the first tree, appearing in an unoccupied space within 5 feet of the second tree. Both trees must be large or bigger.
Disengage (PHB p.192):
If you take the Disengage action, your movement doesn't provoke opportunity attacks for the rest of your turn.