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troperville

The boys grew up, and Esau became a skillful hunter, a man of the open country, while Jacob was content to stay at home among the tents. Isaac, who had a taste for wild game, loved Esau, but Rebekah loved Jacob.

- Genesis 25:27-28

In a family of two or more children, each child has a stronger affinity for a different parent. There are two main ways this can happen:

Each child resembles a different parent in looks or personality.
Each parent is closer to or favours a different child.

These tend to go hand-in-hand, but not always. Sometimes, even when parents strive to be impartial, they still find that each child is more like one parent than the other. Conversely, taking after a parent is no guarantee of being that parent's favourite; it may even have the opposite effect.

May lead to accusations of Parental Favouritism, or to each child feeling like one parent's Unfavourite. Can lead to Sibling Rivalry and even Cain and Abel, particularly when the parents are at odds with each other. May contribute to Middle Child Syndrome if there are more than two children.

On the other hand, a family like this may be perfectly happy if each child is content to be one parent's favourite. And - as so often happens - the children may well be closer to each other than to either parent.

The most common stereotype is a Daddy's Girl and a Momma's Boy. A common variation, especially if both siblings are the same gender, is for the tougher or "manlier" one to be closer to the dad, and the more sensitive or "girlier" one to be closer to the mom.

Examples:

Literature

In A Song of Ice and Fire, the Stark children all physically take after one of their parents. Arya takes after Ned, and Robb, Sansa, Bran, and Rickon all take after Catelyn. Personality-wise, only Robb really takes after either of the parents - he takes after Ned. (Jon both looks and acts like Ned, but isn't Cat's son.)
- This causes Catelyn some amount of angst, at least before other concerns rear their head: Jon is the only son who takes after Ned physically, yet he's the illegitimate one.
Glynn and Ember from Isobelle Carmody's 'Legendsong Saga'. Glynn is much closer to her father while Eber takes after her mother in both looks and talents. It is mentioned that their mother didn't really want children at all, and is only able to love Ember because they are so similar. Strongly influences Glynn's personality.
John Steinbeck's East of Eden relies heavily on the Cain/Abel, Jacob/Esau motif throughout its entirety. There are two sets of brothers: Adam and Charles and then Aron and Cal (sons of Adam and a crazy maniacal whore named Cathy). Adam and Aron = Jacob/Abel, Charles and Cal = Esau/Cain. Adam and Aron are good boys who take after their dads. Charles and Cal are more... unstable and tend to flirt with evil, taking after their mothers.
Lowlands of Scotland Series: Given that the books are based on the story of Jacob, the presence of this trope is unsurprising. Jamie is the more genteel, thoughtful, and sly brother who is close to and favored by their mother; Evan is the rougher, tougher, blunt hunter who is closer to and favored by their father. They have all the Sibling Rivalry of the originals, too.

Live-Action T.V.

Heroes:
- According to Mama Petrelli, Nathan takes after his father but Peter was always her favourite. Of course, Mama Petrelli's a lying bitch who wanted Peter to explode, and her husband's a complete monster who tried to have Nathan murdered. Clearly favouritism is a relative thing among the Petrellis.
- Hiro and his sister take after their mother and father, respectively. They have a lot of tension when it comes to Hiro's hero-ing vs. running the company.
That '70s Show: Red makes no secret of his preference for Laurie, while Kitty is subtler about her favoritism toward Eric.
Revenge: In the Grayson family, Daniel is clearly Victoria's favorite, while Charlotte seems to be Conrad's.
On Roseanne Dan's favorite is Darlene, while the title character is closer to Becky. They spend an episode trying to switch and each form a closer relationship with the other daughter. When everything blows up in their faces, they agree to go back to the way things were.
Dan: So which one of us gets to ruin DJ's life?
Roseanne: We'll flip a coin.
In Modern Family, Mitchell always considered himself closer to his mother, while Claire felt closer to their father.

Mythology & Religion

The prototypical example, at least in western culture, is probably Jacob and Esau from The Bible. They are twins, but Esau, the elder brother, is favoured by his father, while Jacob is his mother's favourite. The latter two even conspire successfully to cheat Esau out of his inheritance. ^*

Theatre

In The Lion In Winter, Henry wants his youngest son, John, to succeed him, but his wife is plotting to get Richard onto the throne. Poor Geoffrey has to make his own luck.

Western Animation

Avatar The Last Airbender: Ozai preferred Azula, Ursa preferred Zuko. Of course, this is a Big Screwed-Up Family being talked about here.
Daria has a subtle example where Daria and Jake seem particularly close while Helen bonds more easily with Quinn. However, Helen in particular still manages to help Daria on several occasions (generally more competently than Jake, if not quite as easy-going).
The Simpsons does this with Bart mirroring Homer's lazier, more irresponsible attitude and Lisa closer to Marge's responsible nature (including nigh-identical disapproving sounds). This is often particularly highlighted in episodes where the parents spend time with the "opposite" child (Bart and Marge or Lisa and Homer)