If my units are at Dreaming Tree and I play a spell that targets my unit but the spell gets countered, do I get the draw from the Dreaming Tree?

Yes, you still get the draw from The Dreaming Tree.

According to the established ruling for this interaction (RiftJudge FAQ #8781):

1. The Dreaming Tree triggers the moment you play a spell that targets a friendly unit at that location. This trigger is placed on the chain above the spell you just played.
2. If your opponent plays a card like Defy to counter your spell, that counter effect is placed on the chain above the draw trigger.
3. When the chain resolves (Last-In, First-Out):
   • The counter resolves first, negating your spell.
   • The Dreaming Tree draw trigger resolves next, allowing you to draw your card.

Because the draw trigger is added to the chain the moment the spell is played, it remains on the chain and resolves regardless of whether the original spell is countered.