What made the Sorting Hat sort Albus Severus into Slytherin?
I’m OK with an answer based on specific traits that match known house characteristics if an unambiguous canon explanation doesn’t exist.
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The text (both Sorcerer's/Philosopher's Stone and Cursed Child) suggests that Albus, like his father, was influenced by the new friend he made on the Hogwarts Express. To see how, look at the similarities between Albus and Harry.
Each was out of place and adrift when he boarded the Hogwarts Express. Harry, because he was new to the Wizarding world, and Albus because he was insecure about who he was.
Each quickly made a friend on the Hogwarts Express who was confident of the house he would be in: Ron was (fairly) confident he would be in Gryffindor, Scorpius in Slytherin.
Each ended up in the same house as his new friend.
Based on these similarities, my theory is that, for each of them, his new friend provided some direction that may have influenced his House placement. In Harry's case, this is clear from his conversation with Ron on the Hogwarts Express:
"What House are your brothers in?" asked Harry.
"Gryffindor," said Ron. Gloom seemed to be settling on him again. "Mum and Dad were in it, too. I don't know what they'll say if I'm not. I don't suppose Ravenclaw would be too bad, but imagine if they put me in Slytherin."
"That's the House Vol-, I mean, You-Know-Who was in?"
"Yeah," said Ron. He flopped back into his seat, looking depressed.
And then later:
[Malfoy] turned back to Harry. "You'll soon find out some wizarding families are much better than others, Potter. You don't want to go making friends with the wrong sort. I can help you there."
He held out his hand to shake Harry's, but Harry didn't take it.
"I think I can tell who the wrong sort are for myself, thanks," he said coolly.
Finally, during the Sorting:
"So where shall I put you?"
Harry gripped the edges of the stool and thought, Not Slytherin, not Slytherin.
"Not Slytherin, eh?" said the small voice. "Are you sure? You could be great, you know, it's all here in your head, and Slytherin will help you on the way to greatness, no doubt about that--no? Well, if you're sure--better be GRYFFINDOR!"
As @Skooba also mentioned in the comments, Hagrid had also told Harry that, "There's not a witch or wizard who went bad who wasn't in Slytherin." I had forgotten this instance of another friend also influencing his choice of House.
The confrontation with Malfoy on the Hogwarts Express is where Harry was truly sorted; the Sorting just confirmed this. Had Ron not implied that Slytherin was not a desirable House to be in, contributing to Harry's reaction to Malfoy and causing the thought "Not Slytherin" to run through his mind, he might well have ended up in Slytherin, due to his ambition.
For example, suppose Harry had met and become good friends with Ernie Macmillan on the Hogwarts Express. Mightn't he have then become a Hufflepuff? Certainly at different points during the books, he exhibited some of the hardworking qualities valued for the Hufflepuffs (although, he probably exhibited un-Hufflepuff qualities more often, the slacker).
Albus had a similar experience on the Hogwarts Express:
ROSE (spotting ALBUS'S loving look at the Chocolate Frogs): Al. We need to concentrate.
ALBUS: Concentrate on what?
ROSE: On who we choose to be friends with. My mum and dad met your dad on their first Hogwarts Express, you know...
ALBUS: So we need to choose now who to be friends with for life? That's quite scary.
And then, Albus makes his choice:
SCORPIUS and ALBUS look at each other and something passes between them.
ROSE: Yes, well, we probably should sit somewhere else. Come on, Albus.
ALBUS is thinking deeply.
ALBUS: No. (Off ROSE'S look.) I'm okay. You go on...
ROSE: Albus. I won't wait.
ALBUS: And I wouldn't expect you to. But I'm staying here.
ROSE looks at him a second and then leaves the compartment.
As with his father, Albus' true sorting occurred here. The Sorting Hat merely confirmed this.
In the Cursed Child, we see an alternate reality in which Albus successfully entered Griffindor in order to spite his friend.
Harry: "Panju bet you couldn't get into Griffindor."
Albus feels sad and misplaced there; not only because he is the Albus we know who belongs in an alternate dimension, but also because, in a school lesson, his peers accuse him of hating it in Griffindor
Polly Chapman: He hates it here, we all know he'd rather be in Slitherin.
Now, taking into account the advice given by Harry-
Harry: The Hat lets you choose. It did for me.
Scorpius could fit into either house. However, he only truly wants to be in one of them.
Finally, it could be argued that Albus had certain traits typical of Slytherins, such as:
Arrogance: believing he could save Cedric by using a time turner. Not only that, but *using an illegal, banned magical item (said time-turner), and believing he could rewrite the past, and do it all better than his father did.
Cunning/slyness: going against his father's back with this.
Scorpius: "Thanks for staying."
Albus: "I'm not staying for you, it's for your sweets."
And, ultimately, Dumbledore said:
"You know, I sometimes think we sort too soon."
slytherin's are the ones who act with taking everything in their minds ( Snape plot twist). Gryffindor's act without judging or going deep in details ( reference to Philosopher's Stone when they suspected snape), hufflepuf's are just simple and tricked by others ( if cerdic was alive he would have become a death eater) whereas ravenclaw's act with logic not emotions (Raven Ravenclaw killed her known mother). (no offence to any house),therefore he is in slytherin
there's a very logical reason behind this.As we know as slytherin's are cunning.If we notice in cursed child Albus depicted it in every scene espically when he figured out delphini's plans after seeing the Panju (Ron and Lavender's son) and Hermonie's state. Therefore this showed his talent of judging the scenario with smartness, something only unique in slytherin
Albus Severus Potter must in the logic of all things Rowling be in house Slytherin. His name is
He is an Asp, and if you have not come across this word before, it is an increasingly obscure term for certain species of snakes common to Europe; sidenote: according to legend Queen Cleopatra committed suicide by enticing an asp to bite her.