12.
Binary Trees
Written by Kelvin Lau
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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 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.