12.
Binary Trees
Written by Kelvin Lau
Heads up... You're reading this book for free, with parts of this chapter shown beyond this point as text.
In the previous chapter, you looked at a basic tree where each node can have many children. A binary tree is a tree where each node has at most two children, often referred to as the left and right children:
Binary trees serve as the basis for many tree structures and algorithms. In this chapter, you’ll build a binary tree and learn about the three most important tree traversal algorithms.
Implementation
Open the starter project for this chapter. Create a new file and name it BinaryNode.swift. Add the following inside this file:
public class BinaryNode<Element> {
public var value: Element
public var leftChild: BinaryNode?
public var rightChild: BinaryNode?
public init(value: Element) {
self.value = value
}
}
In the main playground page, add the following:
var tree: BinaryNode<Int> = {
let zero = BinaryNode(value: 0)
let one = BinaryNode(value: 1)
let five = BinaryNode(value: 5)
let seven = BinaryNode(value: 7)
let eight = BinaryNode(value: 8)
let nine = BinaryNode(value: 9)
seven.leftChild = one
one.leftChild = zero
one.rightChild = five
seven.rightChild = nine
nine.leftChild = eight
return seven
}()
This code defines the following tree by executing the closure:
Building a diagram
Building a mental model of a data structure can be quite helpful in learning how it works. To that end, you’ll implement a reusable algorithm that helps visualize a binary tree in the console.
extension BinaryNode: CustomStringConvertible {
public var description: String {
diagram(for: self)
}
private func diagram(for node: BinaryNode?,
_ top: String = "",
_ root: String = "",
_ bottom: String = "") -> String {
guard let node = node else {
return root + "nil\n"
}
if node.leftChild == nil && node.rightChild == nil {
return root + "\(node.value)\n"
}
return diagram(for: node.rightChild,
top + " ", top + "┌──", top + "│ ")
+ root + "\(node.value)\n"
+ diagram(for: node.leftChild,
bottom + "│ ", bottom + "└──", bottom + " ")
}
}
example(of: "tree diagram") {
print(tree)
}
---Example of tree diagram---
┌──nil
┌──9
│ └──8
7
│ ┌──5
└──1
└──0
Traversal algorithms
Previously, you looked at a level-order traversal of a tree. With a few tweaks, you can make this algorithm work for binary trees as well. However, instead of re-implementing level-order traversal, you’ll look at three traversal algorithms for binary trees: in-order, pre-order and post-order traversals.
In-order traversal
In-order traversal visits the nodes of a binary tree in the following order, starting from the root node:
extension BinaryNode {
public func traverseInOrder(visit: (Element) -> Void) {
leftChild?.traverseInOrder(visit: visit)
visit(value)
rightChild?.traverseInOrder(visit: visit)
}
}
example(of: "in-order traversal") {
tree.traverseInOrder { print($0) }
}
---Example of in-order traversal---
0
1
5
7
8
9
Pre-order traversal
Pre-order traversal always visits the current node first, then recursively visits the left and right child:
public func traversePreOrder(visit: (Element) -> Void) {
visit(value)
leftChild?.traversePreOrder(visit: visit)
rightChild?.traversePreOrder(visit: visit)
}
example(of: "pre-order traversal") {
tree.traversePreOrder { print($0) }
}
---Example of pre-order traversal---
7
1
0
5
9
8
Post-order traversal
Post-order traversal only visits the current node after the left and right child have been visited recursively.
public func traversePostOrder(visit: (Element) -> Void) {
leftChild?.traversePostOrder(visit: visit)
rightChild?.traversePostOrder(visit: visit)
visit(value)
}
example(of: "post-order traversal") {
tree.traversePostOrder { print($0) }
}
---Example of post-order traversal---
0
5
1
8
9
7
Key points
- The binary tree is the foundation of some of the most important tree structures. The binary search tree and AVL tree are binary trees that impose restrictions on the insertion/deletion behaviors.
- In-order, pre-order and post-order traversals aren’t just important only for the binary tree; if you’re processing data in any tree, you’ll use these traversals regularly.