2-3-4 Trees

2-3-4 Trees: Introduction

• Motivation: Keep a search tree balanced


• Basic ideas:
• Keep 1, 2, or 3 values per node
• Values in nodes and subtrees are ordered (as in BST)
• Split nodes as needed, pushing values up
• All leaves are the same depth


• 2-3 trees invented by John Hopcroft (1970)

2-3-4 Trees: Definition

• Each node in a 234 tree is either a 2-node, 3-node, or 4-node
• 2-nodes have 2 children
• 3-nodes have 3 children
• 4-nodes have 4 children


• The number of children can change


• The number of children depends on the number of values in the node:
• 2-nodes have 1 value
• 3-nodes have 2 values
• 4-nodes have 3 values


• The values and children maintain an ordering (like a binary search tree)


• AKA (2-4) trees

2-3-4 Trees: Example

• Examples (each key is a single letter):
• Node with 2 values and no children: ER
• Node with 3 values and no children: EIR
• Node with 2 values and three children: ER/AAC;HIN;S
• Node with 3 values and four children: EIR/AAC;GH;N;S

2-3-4 Trees: Finding Nodes

• 2-node: left ≤ key; key ≤ right
• 3-node: left ≤ lkey; lkey ≤ middle ≤ rkey; rkey ≤ right
• 4-node: left ≤ lkey; lkey ≤ lmiddle ≤ mkey; mkey ≤ rmiddle ≤ rkey; rkey ≤ right


• How do we search for a value?


• Notice: duplicates can appear in either subtree

2-3-4 Trees: Adding Values

• Traverse tree from top to bottom


• Split any 3-nodes by moving up middle value
• Middle value will always have a place to go. Why?


• If new value is same as node value, go right


• Only add values in bottom nodes

2-3-4 Trees: Example Adding Values

• Let's add A S E A R C H I N G E X A M P L E to a tree
• Empty ; add A
• A ; add S
• AS ; add E
• AES ; add A
• E / AA, S ; add R, C, H
• E / AAC, HRS ; add I, N
• ER / AAC, HIN, S ; add G
• EIR / AAC, GH, N, S ; add E
• I / E, R / AAC, EGH, N, S ; add X, A, M, P, L, E
• I / AEG, NR / A, AC, EE, H; LM, P, SX

Why the Tree Remains Balanced

• Height only grows when split root
• Thus height of all nodes is increased at same time


• Thus, tree remains balanced!

2-3-4 Tree Implementation

• Having a variable number of values and pointers per node complicates implementation


• Red-Black Trees are a binary tree implementation of Red Black Trees
• Every node has a color: red or black
• A red node is part of its parent in the equivalent 2-3-4 tree
• RB trees allows using a simpler data, but the implementation is more complicated

2-3-4 Trees and Other Balanced Trees

• We looked at a top-down tree; bottom up trees also exist


• Can also create 2-3 trees


• 2-3-4 Trees are a kind of B-Tree
• B-Trees are used in storing databases on disk