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Data Structures & Algorithms in Swift

Fourth Edition · iOS 15 · Swift 5.5 · Xcode 13

17. AVL Tree Challenges
Written by Kelvin Lau

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Here are three challenges that revolve around AVL trees. Solve these to make sure you have the concepts down.

Challenge 1: Number of leaves

How many leaf nodes are there in a perfectly balanced tree of height 3? What about a perfectly balanced tree of height h?

Challenge 2: Number of nodes

How many nodes are there in a perfectly balanced tree of height 3? What about a perfectly balanced tree of height h?

Challenge 3: A tree traversal protocol

Since there are many variants of binary trees, it makes sense to group shared functionality in a protocol. The traversal methods are a good candidate for this.

Solutions

Solution to Challenge 1

A perfectly balanced tree is a tree where all the leaves are in the same level, and that level is completely filled:

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func leafNodes(inTreeOfHeight height: Int) -> Int {
  Int(pow(2.0, Double(height)))
}

Solution to Challenge 2

Since the tree is perfectly balanced, the number of nodes in a perfectly balanced tree of height 3 can be expressed by the following:

func nodes(inTreeOfHeight height: Int) -> Int {
  var totalHeight = 0
  for currentHeight in 0...height {
    totalHeight += Int(pow(2.0, Double(currentHeight)))
  }
  return totalHeight
}
func nodes(inTreeOfHeight height: Int) -> Int {
  Int(pow(2.0, Double(height + 1))) - 1
}

Solution to Challenge 3

First, create the following protocol:

protocol TraversableBinaryNode {

  associatedtype Element
  var value: Element { get }
  var leftChild: Self? { get }
  var rightChild: Self? { get }
  func traverseInOrder(visit: (Element) -> Void)
  func traversePreOrder(visit: (Element) -> Void)
  func traversePostOrder(visit: (Element) -> Void)
}
extension TraversableBinaryNode {

  func traverseInOrder(visit: (Element) -> Void) {
    leftChild?.traverseInOrder(visit: visit)
    visit(value)
    rightChild?.traverseInOrder(visit: visit)
  }

  func traversePreOrder(visit: (Element) -> Void) {
    visit(value)
    leftChild?.traversePreOrder(visit: visit)
    rightChild?.traversePreOrder(visit: visit)
  }

  func traversePostOrder(visit: (Element) -> Void) {
    leftChild?.traversePostOrder(visit: visit)
    rightChild?.traversePostOrder(visit: visit)
    visit(value)
  }
}
public final class AVLNode<Element>
extension AVLNode: TraversableBinaryNode {}

example(of: "using TraversableBinaryNode") {
  var tree = AVLTree<Int>()
  for i in 0..<15 {
    tree.insert(i)
  }
  tree.root?.traverseInOrder { print($0) }
}
---Example of: using TraversableBinaryNode---
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