2. We ll add new nodes to the AVL as leaves just like we did for Binary Search Trees (BSTs). a) Add the key 90 to the tree?

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1 eam #: bsent:. n VL ree is a special type of inary Search ree (S) that it is balanced. y balanced I mean that the of every s left and right subtrees differ by at most one. his is enough to guarantee that a VL tree with n s has a no worst than Θ( log n ). herefore, insertions, deletions, and search are in the worst case Θ( log n ). n example of an VL tree with integer keys is shown below. he of each is shown a) Label each with one of the following balance factors: EQ if its left and right subtrees are the same L if its left subtree is one taller than its right subtree R if its right subtree is one taller than its left subtree Each VL tree usually maintains a balance factor in addition to the key, payload, etc.. We ll add s to the VL as leaves just like we did for inary Search rees (Ss). a) dd the key 90 to the tree? b) Identify the closest to the inserted (90) that no longer satisfies the balanced property of an VL tree. his is called the pivot. c) onsider the subtree whose root is the pivot. How could we rearrange this subtree to restore the VL balanced property of VL tree? (Draw the tree resulting tree below). ypically, an insertion of a key into an VL requires the following steps: compare the key with the current tree s key (as we did in the put method in the S) to determine whether to recursively insert/put the key into the left or right subtree add the key as a leaf as the base case(s) to the recursion as the recursion unwinds (i.e., after you return from the recursive call) adjust the balance factors of the s on the search path from the back up to the root of the tree. o aid in adjusting the balance factors, will modify the insertiong/put method so that it returns rue if the subtree got taller and False otherwise. as the recursion unwinds if we encounter a pivot (as in question above) we perform one or two rotations to restore the VL tree s -balanced property. Lecture 9 Page

2 eam #: bsent: For example, consider the previous example of adding 90 to the VL tree. efore the insertion the pivot was already R (tall right - right subtree had a one greater than its left subtree). fter inserting 90, the pivot s right subtree had a more than its left subtree which violates the VL tree s -balance property. his problem is handled with a left rotation about the pivot as shown in the following generalized diagram: efore the insertion: fter the insertion, but before rotation: 'R' 'R' Rotate Left at Pivot Recursion has unwound to pivot and indicated a taller subtree! 'R' 's balance factor was already adjusted before pivot found n fter left rotation at pivot: n a) ssuming the same initial VL tree ( is R) if the would have increased the of, would a left rotation about the have rebalanced the VL tree? Lecture 9 Page

3 eam #: bsent: 4. efore the insertion if the pivot was already R (tall right - right subtree had a one greater than its left subtree) and if the is inserted into the left subtree of the right child, then we must do two rotations to restore the VL-tree s -balance property. efore the insertion: 'R' fter the insertion, but before first rotation: 'R' st Rotate 'R' Right at Node L R L R n - n - n - Recursion has unwound to pivot and indicated a taller subtree! 's & 's balance factors have already been adjusted before 'L' the pivot was found fter the left rotation at pivot and balance factors adjusted correctly: fter right rotation at, but before left rotation at pivot: 'R' 'L' L n - R Rotate Left at Pivot L n - 'R' R 'L' a) Suppose that the was inserted into the right subtree of the pivot s right child, i.e., inserted in L instead R, then the same two rotations would restore the VL-tree s -balance property. However, what should the balance factors of s,, and be after the rotations? Lecture 9 Page

4 eam #: bsent: 5. Now let s consider the partial VL tree code: class reenode: def init (self,key,val,parent=none,left=none,right=none, bal=): self.key = key self.payload = val self.balance = bal self.lefthild = left self.righthild = right self.parent = parent def rotateright(self): Self = reenode(self.key, self.payload, self, self.lefthild.righthild, self.righthild,self.balance) self.key = self.lefthild.key self.payload = self.lefthild.payload self.balance = self.lefthild.balance self.righthild = Self self.lefthild = self.lefthild.lefthild if Self.lefthild: Self.lefthild.parent = Self if Self.righthild: Self.righthild.parent = Self if self.lefthild: self.lefthild.parent = self def put(self,key,val): if key < self.key: <**** ODE OMIED -- O E OMPLEED OMORROW S L 9 ****> else: # key > self.key so add key to right subtree if self.righthild: tallerrightsubtree = self.righthild.put(key,val) if tallerrightsubtree: # Here we check to see if the subtree got taller if self.balance == 'L': self.balance = return False elif self.balance == : self.balance = 'R' return rue else: # wo too tall on right now if self.righthild.balance == 'R': # only rotate left at pivot self.balance = # preset balance factors to self.righthild.balance = # be correct after pivoting else: # need to rotate right at, then rotate left at pivot if self.righthild.lefthild.balance == 'R': self.balance = 'L' self.righthild.balance = elif self.righthild.lefthild.balance == 'L': self.balance = self.righthild.balance = 'R' else: self.balance = self.righthild.balance = self.righthild.lefthild.balance = self.righthild.rotateright() self.rotateleft() return False else: # base case in search -- add as a leaf self.righthild = reenode(key,val,self) if self.balance == : self.balance = 'R' return rue else: self.balance = return False a) Where in the above code do the balance factors get set for your answer to question 4 (a)? Lecture 9 Page 4

5 eam #: bsent: 6. omplete the below figure which is a mirror image to the figure in question, i.e., inserting into the pivot s left child s left subtree. Include correct balance factors after the rotation. efore the insertion: fter the insertion, but before rotation: 'L' 'L' n 'L' Rotate Right at Pivot fter right rotation at pivot: Lecture 9 Page 5

6 eam #: bsent: 7. omplete the below figure which is a mirror image to the figure in question 4, i.e., inserting into the pivot s left child s right subtree. Include correct balance factors after the rotation. efore the insertion: fter the insertion, but before first rotation: 'R' 'R' 'L' L R L R n - n - n - 'R' fter the rightt rotation at pivot and balance factors adjusted correctly: fter left rotation at, but before right rotation at pivot: Lecture 9 Page 6