Trees Continued

Week 8, Wednesday

February 25, 2026

3-Letter Jotto: Let’s Play

The setup:

DEW  FOX  GEL  INK
JOG  MOB  NAP  NIP
NOR  NUT  OAT  OIL
PUG  SOY  WOO  YAK

• There are 16 possible secret words – you choose one.
• I guess a word; You respond with the count of letters in common (0, 1, or 2). If I get 3, the game is over!
• I will continue to guess until I guess your word.

The Play: Turn and face the back of the room.

My Strategy

What Strategy Did You Discover?

Discussion questions:

• What makes a “good” first guess?
• What happens if you guess NAP, then NIP, then NUT?
• Can you always win in 2 guesses? Why or why not?

Lead into: the strategy IS a tree! Each guess partitions the remaining words by their Jotto score. Good guesses split evenly. Bad guesses leave big partitions.

Lower bound discussion: Each guess has 3 branching outcomes (0, 1, or 2 matches). Getting 3 matches means you guessed correctly—game over, no more branches needed! So with 3 outcomes per guess and 16 words, theoretically log₃(16) ≈ 2.5 guesses should suffice. Since we can’t make half a guess, the lower bound is 3 guesses. But even that’s hard to achieve in practice!

Why Trees?

Trees appear everywhere in computer science:

Domain	Tree Structure
File systems	Directories contain subdirectories
HTML/DOM	Elements contain child elements
Decision making	Each choice leads to sub-choices
Classification	“Is it X? If yes, go left…”
Game AI	Possible moves branch into more moves

Trees: The Most Important Nonlinear Structure

“Trees are the most important nonlinear structure in computer science.”

— Donald Knuth, The Art of Computer Programming

What’s special about this structure? ________________________

Tree Terminology

• One of the vertices is called the root of the tree. Guess which one it is.
• Make an English word containing the names of the vertices that have a parent but no sibling.
• How many parents does each vertex have?
• Which vertex has the fewest children?
• Which vertex has the most ancestors? descendants?
• What is d’s depth? What is d’s height?
• List all the vertices in b’s left subtree.
• List all the leaves in the tree.

Tree Vocabulary (reference)

• root: the single node with no parent
• leaf: a node with no children
• child: a node pointed to by me
• parent: the node that points to me
• sibling: another child of my parent
• ancestor: my parent or my parent’s ancestor
• descendant: my child or my child’s descendant
• subtree: a node and its descendants
• depth of node x: number of edges on path from root to x
• height of node x: number of edges on longest path from x to a leaf

d-ary Tree Definition

A d-ary tree T is either:

• _______________________________________

OR

• _______________________________________

Fill in via annotation: - Empty (no nodes) - A root node with up to d subtrees (each a d-ary tree)

Binary tree = d-ary tree with d=2

The tree shown has d=3 (ternary). Each subtree of the root is colored differently to emphasize the recursive structure.

Special Trees

Complete Tree

Full Tree

Perfect Tree

This is a matching activity. Students should draw lines connecting: - Perfect Tree: All levels completely filled (coral tree, top) - this is both full AND complete - Complete Tree: All levels filled except possibly the last, which is filled left-to-right (teal tree, middle) - Full Tree: Every node has 0 or 2 children (yellow tree, bottom) - note it can be unbalanced!

Key insight: Perfect ⊂ Complete, and Perfect ⊂ Full, but Complete and Full are independent properties.

Week 8 Summary

• Trees are hierarchical structures with a root, children, and leaves
• Binary trees have at most 2 children per node
• Trees are recursively defined: a tree is empty OR a root with subtrees
• Tree terminology: root, leaf, parent, child, sibling, ancestor, descendant, depth, height
• Decision trees model sequences of choices