What connections can be made between the woods of Sondheim's Into the Woods and those of A Midsummer Night's Dream?
In both the play Into the Woods and A Midsummer Night's Dream, the woods represent both a magical place for the fulfillment of wishes but also a place where magic gets out of hand and creates mayhem.
In the opening act in Into the Woods, each main character makes his or her way into the woods to fulfill either a wish or a need. For, example the Baker goes into the woods after learning he and his wife are childless as a result of a curse from the witch; specifically, he goes into the woods to collect the items the witch says will break the curse. Cinderella goes into the woods to cry at her mother's grave and tell her mother of her wish to go to the festival, Jack goes into the woods to travel to market to sell his cow. The initial results of the characters traveling into the woods is that the each get, in general, what he or she wants. Even Little Red Riding Hood learns a great lesson, and Jack sells his cow to the Baker for some magic beans. At first, every one is content with what each character has either gained or learned. However, Jack's magic beans of course release a giant who kills many of the characters in the second act. Once the remaining characters have killed the giant, all of the remaining characters, as well as the deceased characters, sing about the lessons they have learned. Hence, for Into the Woods, the woods represent a fulfillment of desires or wishes, but the fulfillment also leads to more severe consequences.
Similarly, in A Midsummer Night's Dream, the escape into the woods creates both a dream-like as well as a nightmarish situation for the characters. Hermia and Lysander venture into the woods to be married in a neighboring city, as well as to escape Hermia's father's decree that she must marry Demetrius and the Athenian punishment if she does not. Demetrius ventures into the woods to pursue Hermia and challenge Lysander for Hermia, while Helena goes into the woods to pursue Demetrius. Puck sees Demetrius rejecting Helena and uses magic to give her exactly what she wants, which is Demetrius's love. Then things get even further mixed up when Puck again uses magic on Lysander who also falls in love with Helena. So, now, Helena has the situation of her dreams--two men in love with her when both scorned her before. But she is too shocked to believe in the reality of the situation and sees it as a nightmare instead because she believes that all three characters, Hermia, Lysander, and Demetrius, are mocking her. Hence, in this respect, Helena has also journeyed into the woods to have her wishes fulfilled but encounters what she thinks is a nightmarish situation instead. We especially see Helena interpret what could be a dream situation for her as a nightmare in the lines:
O spite! O hell! I see you all are bent
To set against me for your merriment.
If you were civil and knew courtesy,
You would not do me thus much injury. (III.ii.147-50)
Just like Helena, Hermia had also journeyed into the woods to have her wish of marrying Lysander fulfilled but has also encountered a nightmare instead because now both men who were in love with her now both scorn her.
Therefore, we see that in both plays, the woods represent a magical place where wishes are meant to be fulfilled, but the magic actually wreaks havoc instead, creating a nightmarish situation.